help-0

[vale wigeon]

don't thank me when i haven't even begun to help.

did you think you'd just get the answer handed to you here?

[jovial mica]

no, he will spend quite a while explaining how to solve questions like thatand ask me to try a few questions.

but he won't do that question, he will do another question to show us how and then make us do that question.

[stray shard]

Why are linear differential equations called so?
According to wiki

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form

a_0 y + a_1 y' + ... + a_n y^(n) = 0

But why is this form linear? Is it because their solutions follow the rule Sum of any two solutions is another solution ?
I've another naive doubt, the solution of
dy/dx = y, is y = Ce^x
Is e^x linear, no right?

[vale wigeon]

"linear" doesn't mean that any of its solutions has to be of the form y = ax + b

if anything, it's called that way as an appeal to linear algebra: the map $L(y) = a_0 y + a_1 y' + \dots + a_n y^{(n)}$ is a linear operator: for any two constants $c_1, c_2$ and functions $y_1, y_2$ you have $$L(c_1y_1 + c_2y_2) = c_1 L(y_1) + c_2 L(y_2)$$

this has many profound consequences, including the solution set of a homogeneous DE being closed under addition [and also scaling]

[ocean seal]

Ann

[stray shard]

Okay cool, thank you

[split fulcrum]

no idea how to find m or do part b for this problem

[split fulcrum]

<@&286206848099549185>

[plucky crow]

Given the fact that dy/dx is speed

And dy/dx = dy/dt * dt/dx

It should be fine? Or am I wrong

[split fulcrum]

I'm confused

how do i get dt/dx?

I found the speed at t = 3, still not sure how to find m

[manic quail]

With mean, they basically mean average, median is just the "middleest" number, and mode is the number that comes most often.

[vale wigeon]

the median is the point that splits your data set in half by point count

i.e. half below half above

[sturdy fog]

why is square root of y^6 y^3?

[vale wigeon]

because (y^3)^2 = y^6.

[alpine sable]

hi

can someone help?

can i get help

@glass lichen my saviour!

wait what it fromatted

[glass lichen]

what?

[alpine sable]

can you help me please if possible

[alpine sable]

with this

[glass lichen]

no

[alpine sable]

understandable

can anyone please correct me if i am wrong ?

please, explanation is very much welcomed

can i get help i know this is basic math but my braincells about to pop

[sturdy fog]

ohhh thank you @vale wigeon

[alpine sable]

for 1 i got positive infinity

for 2 im a bit confused

[vale wigeon]

do you know what the graph of y=tan(x) looks like?

alternatively you could write tan(θ) as sin(θ)/cos(θ)

[alpine sable]

@vale wigeon i did and i got 1/0 which i think is positive ifninity

[vale wigeon]

you need to be very careful with 1/0

in this case it is positive infinity because cos(x) approaches 0 from above as x goes to pi/2 from below

[alpine sable]

understood

and for d i got DNE

[vale wigeon]

that is correct.

[alpine sable]

since from the left its -1 and from the right positive 1

awesome

does anyone know Statistics problem solving?

[alpine sable]

@vale wigeon what about this one

IK that x cant be 3

[vale wigeon]

?

what are you asked for

[alpine sable]

I need to find points where f is discontinous

[vale wigeon]

oh

[alpine sable]

sorry the picture cropped wrong

[vale wigeon]

3 isnt even in the domain

[alpine sable]

yeah so its discontinous there right?

since it never approaches 3

[glass lichen]

it approaches 3, it's just an asymptote at x=3

[alpine sable]

yeah

so isnt that a sign of discontinuity?

[glass lichen]

you just need to see if $\lim_{x\to 5}f(x)=4$

[ocean seal]

moshill1

[glass lichen]

to determine if that point is a jump or not

[alpine sable]

well

as x approaches 5- the lim = 2

as x approaches 5+ lim is also 2

however at 5, y = 4

[glass lichen]

yes, so it's jump discontinuous

[alpine sable]

but why is that the only discontinuity

why isnt x = 3 a discontinuit?

[glass lichen]

it is

No one said it wasnt

[alpine sable]

so what should i say for that

since it isnt a concrete point?

asymptote discontuity?

[glass lichen]

yes... it's an infinite discontinutiy / vertical asymptote

[alpine sable]

got it

sorry if i am frustrating you in any way, im just trying to understand

this one im uite stumped

i cant set them equal to each other

and neither can i subsitiute anything

[alpine sable]

you want ax^2+bx to go through the points (1,4) and (2,10)

yes but how do i get it

[alpine sable]

when x=1, the value of ax^2+bx is 4
when x=2, the value of ax^2+bx is 10
can you use this to make a system of equations and solve for a and b?

@alpine sable but they're not equal to anything

i have a+b and 4a+2b

[alpine sable]

when x=1, the value of ax^2+bx is 4
you're not using all the information given

a(1)^2+b(1)=4

do the same thing for the second bit

i got 1 and 3 for a and b respectively

so x^2+3x?

@alpine sable

[alpine sable]

correct

@alpine sable

im having troube with c and d

[kindred helm]

hey guys i got a circle like this. the task is to figure out the alpha angle with out any cas calculators but i did it on geogebra. how can you do it neatly on paper

[alpine sable]

f(x) is continuous, use IVT

[alpine sable]

repost that in an unoccupied channel

[alpine sable]

yup did that

for b i got y+1 = 8x-8

derivative = slope of tangent line

for c i wanna say its decreasing at x = 11 since slope of tangent line is negative

[alpine sable]

that's correct

rate of change is negative

got it

and for the last one

[alpine sable]

i wanna say its at x = 5

since f'(5) = 0

[alpine sable]

when the derivative is 0 the function is neither increasing or decreasing

got it

@alpine sable finally, i have this monstrosity

discontinuity at x=5

also functions won't be differentiable at "sharp points" like this

[analog terrace]

can someone give me the gist on how to answer this question?

In how many ways can a committee of 6 persons be chosen from 10 single ladies and 8 men if there must be 2 men and 4 ladies?

i know how to solve combinations but i dont know how if there's more than two variables

[alpine sable]

we have to choose 2 men out of the group of 8 men and we have to choose 4 women out of the group of 10 women

[analog terrace]

oh so i get the number of combinations from each gender then multiply or add?

[alpine sable]

you multiply

because if you take one specific combination of two men you have (10C4) ways to choose the women

[analog terrace]

oh okay. thanks

[alpine sable]

so since we can choose the men in (8C2) ways and each way to choose the men gives us (10C4) ways to choose the women, we would multiply

[willow bluff]

Does anyone know the answer to this?

[modest crow]

pythagoras

[pearl wraith]

no

[modest crow]

idk the mathematician but

hypotenus AC=18+r

[pearl wraith]

use the 30 and 18 to find A angle

[modest crow]

they want the radius, why you making it complex

[pearl wraith]

then using thatangle you can find r

[alpine sable]

you don't need trig

just use pythagorean theorem

[modest crow]

you cant find the angle with AC=18+r, homie

it gets complicated

[analog terrace]

use pythagorean theorem. the opposite side formula

@alpine sable my answer is 5880. am i right?

[alpine sable]

no that's not right

[alpine sable]

oh wait this is for a different question

i though you were answering the geometry question

[analog terrace]

oh lol

[alpine sable]

yes 5880 is right

[analog terrace]

nice

[lean nest]

Suppose I have a bunch of 2D points and I'd like to find 5 of them where (x1 + x2 + x3 + x4 + x5) * (y1 + y2 + y3 + y4 + y5) gives the maximum value, what's an efficient algorithm to do it?

If one exists, is it possible to extend to 3D points?

[ionic jewel]

you have many points and you want to choose 5 of them?

[lean nest]

Correct, choose the 5 of them that will produces the highest value of sum of xs * sum of ys

[ionic jewel]

i dont have a proof but im fairly sure picking the 5 with highest x*y will give you the highest sum of xs * sum of ys

ah perhaps not i see a case im missing

[lean nest]

Yeah, it's like diminishing return, say if the sum of xs are pretty large already a point with large y would be better.

I do have an algorithm that performs decently well but it's more or less brute force with clever tricks to eliminate cases, and it doesn't scale if it's 3D points instead of 2D.

I thought I'd ask and see how this problem could be approached from math angle rather than CS.

[ionic jewel]

alr burrito mind if i ping you later if i figure it out

[lean nest]

I would appreciate that, I don't check here too often.

Alternatively, if you guys know how I would Google this question, I'm guessing there has to be a solution already

This seems to be a question that's very applicable in many fields to optimize things.

[ionic jewel]

i do think my original idea would be quick and close to optimal tho

I'm pretty sure you would have to go out of your way to find a case it doesn't cover

[alpine sable]

yo can sum1 help me with a equation?

[lean nest]

I don't think it's as rare, I think it should be fairly easy to come up with a counter example without too much trouble.

Here's one, if you pick first 4 points plus 5th you will get 81809, but first 4 with 6th you will get 83408 which is better, despite 6th point has lower individual product than 5th.

[alpine sable]

i'd start with the simpler case where you choose 2 or 3 points to maximize the quantity

[lean nest]

Good idea

I haven't done math in a really long time, time to pull out pen and paper.

Though I'm not hopeful in solving this, feels a bit out of my knowledge.

[ionic jewel]

yeah I'm not entirely sure how to even solve the 2 case

[alpine sable]

was this a textbook/hw or competition problem or something you came up with on your own?

[lean nest]

No, this is a tool that I'm coding, so real world application.

[alpine sable]

https://artofproblemsolving.com/articles/files/MildorfInequalities.pdf
maybe something here might help? i haven't made any progress with the question so far

this might help

https://artofproblemsolving.com/wiki/index.php/Root-Mean_Square-Arithmetic_Mean-Geometric_Mean-Harmonic_mean_Inequality

this'll give us an upper bound but we'd still need to show that the upper bound is possible

idk how you'd code an algorithm to return the set of points that'd get you closest to the optimal value except just guess-and-checking

[lean nest]

Brute force is pretty much how the current algorithm operates, so even if there's a way to determine which points would not be useful at all that would still massively reduce the search space and improve algorithm performance.

Because some searches are like "pick 5 from 1200 points" and it takes sometimes up to 20 seconds to compute, which is fast but nowhere near ideal.

[alpine sable]

i might have an idea for how to optimize it

assign each point (x,y) a sum value which would be equal to x+y

and we would only pick from the points that tie in the top 5 ranks for the sum value

wait idk if that would work

[lean nest]

Hmm, I don't think that works

If x overshadows y in magnitude, then that would definitely be wrong.

[alpine sable]

well we can definitely eliminate all the points that have an x and a y that are both less than or equal to the x and y of some other point

so for example if we have (6,5) and (6,4) and a bunch of other points then choosing (6,4) wouldn't be optimal

[lean nest]

I think you would run into the same problem as this

Assigning a "value" for individual points doesn't feel right to me, because the value of a point is also dependent on the value of other previously chosen points.

[alpine sable]

no i'm talking about points where the x and the y are both less than or equal to the x and a y for another point

[lean nest]

Oh then sure yeah.

Hmm, well you could only eliminate them if you get to finding the 5th though, no?

[alpine sable]

true if you have (2,3) (2,2) and 3 (1,1)'s you couldn't just eliminate the (2,2) straight off the bat

[lean nest]

Yep, and also two groups with distinct characteristics.

Say one group has higher xs and another group has higher ys

Even though some points in their respective groups are clearly worse than others, you can't eliminate them.

[alpine sable]

Anyone cam correct me?

[alpine sable]

i might have an idea
start with all the points in group 1
if point A in the group is (x1,y1) and some arbitrary point B in the group is (x2,y2) and either of (x1<y1 and y1<=y2) or (x1<=y1 and y1<y2) is true then we call point A non-optimal
remove all non-optimal points from group 1 and put them into group 2
rinse and repeat, put all non-optimal points from group 2 into group 3 and so on until you finish
first guess-and-check by choosing random combos of 5 numbers from group 1
if you don't have enough points in group 1 then you'd guess-and-check by choosing random combos of 5 numbers from group 1 and group 2, and so on
i think this would save time for large sets because hopefully a lot of the numbers would be removed from group 1, and for large sets we would probably have more than 5 points to choose from in group 1 so the pool of numbers you would guess-and-check from would be a lot smaller

but the code for placing everything into groups might take even more time to run

¯_(ツ)_/¯

[alpine sable]

especially for large sets of points

[lean nest]

Hmm, I'm not quite seeing how it reduces search space

[alpine sable]

yeah because you'd have to compare a point to all the other points to see if it's non-optimal

i dont think it'd help

[lean nest]

Yeah the goal is to reduce search space as much as possible, because 1000 choose 5 is like 10 to the 12th power

The currently algorithm does a really good job at reducing search space most of the time to only a few hundred thousand (kind of arbitrary without knowing the dataset but just to give an idea)

Otherwise that amount of calculation is not possible even with all the computing power in the world.

[ionic jewel]

what runtime is acceptable?

not factorial appearently lmao

[fossil sedge]

Hi can someone help me solve this simultaneous equation

to express x_i as a function of p.5 and p.75

also this

[lean nest]

I'm not even sure, but hey comparison sorts are O(nlogn) so it's not that bad.

I just feel like this optimization problem should be very common in lots of fields, and someone has to have already come up with a good solution, and I probably just don't know the keyword to search for.

[fresh quail]

Hello, can someone help me with GCF and LCM? So GCF=greatest common factor, so for example, 36=2²×3² and 44=2²×11 then the GCF is 2² (it's the common and the greatest) is this correct?

and for LCM=Least common multiple, for the same example, 36=2²×3² and 44=2²×11. can we say in other words for the LCM, it's all the different and greatest? so in this case,LCM= 2²×3²×11. Am I right?

I've really been struggling, help would be so appreciated🥰

[ionic jewel]

that is correct

for both

[fresh quail]

the way of finding the LCM too?

[ionic jewel]

LCM you need all the unique ones

[fresh quail]

and the greatest?

[ionic jewel]

im not sure specifically what you mean with your wording but your answer is right

[fresh quail]

so if we had 2³ and 2², we pick 2³

[ionic jewel]

oh

yes

[fresh quail]

alrightt

thank youu

🥰 🥰

[ionic jewel]

yw

[next pulsar]

Is it enough to solve q(x)=0 and show none of the solutions lie within Z_2 in order to prove it?

[sly rose]

Yes but that would be more trouble than needed

[next pulsar]

Well my first instinct was to just plug in 0 and 1 and show both times q=1

Is that really all?

[sly rose]

Almost all

You also need to try it for -1

[next pulsar]

huh? Now I’m confused, I swear -1 = 1 (mod 2)

[sly rose]

There is no need for mod 2

You just use the fact that if g(a) is positive and g(a+1) is negative or vice versa than there is at least a solution that is not an integer (a is an integer)

Well ofc you can use mod 2

To show that it’s always odd

[next pulsar]

Still don’t understand why I’m allowed to input a number not in Z_2 into q tho

[sly rose]

Polimomials are basically functions defined in In the complex number (unless the problem tells you it’s not)

[tacit olive]

do anyone have chegg

[wild nest]

anyone in here?

[stiff vortex]

try not to get 904.77

it's apparently not correct

[ionic jewel]

36pi

[stiff vortex]

6 is the diameter.. shit

[ionic jewel]

yep

[barren palm]

how would i arrange the -2x bit to get |x-c| on its own

[shell widget]

you can get |x+c| idk about |x-c|

| -2|x| + 2|c| | = | 2|c| - 2|x| | = | |2c| - |2x| | <= |2c + 2x| = 2|x+c|

by reverse triangle inequality

[sly mantle]

use reverse triangle the way it's usually stated

||x|-|y||=<|x-y|

[shell widget]

oh is it both?

| |x| - |y| | <= |x-y| and |x+y|

?

oh it is nvm

[sly mantle]

the ineq w/ |x+y| is from applying triangle again

[shell widget]

I see

Ty

[latent mist]

how come when you differentiate (5x-3)^3 its 15(5x-3)^2 and not 3(5x-3)^2

[sly mantle]

actually no it's not but it still stands

[latent mist]

oh

sry if i interrupted

[sly mantle]

not you

[latent mist]

.. obviously

[sly mantle]

use chain, and don't use x for multiplication if x means a variable

[latent mist]

x is a variable

yes

didnt mean to put that x there

but the book im using didnt use chain rule to get that

[shell widget]

Derivative of (ax + b)^n is n(ax+b)^(n-1) * [d/dx (ax+b)]

This is power rule + chain rule

[latent mist]

ok

i see

thank you

[plucky crow]

@shell widget

can u help in question 2 if u have the time pls

[shell widget]

if i know it

sure

[crisp pollen]

is 1 parrell?

and im not sure for 2

[meager patio]

so

[plucky crow]

@crisp pollen factor out -2/3 from v1

[strange raft]

may i ask a question or will be intrupting

[ionic jewel]

$\frac 1n \sum_{i=0}^{n} i$

[ocean seal]

bunny

[ionic jewel]

@strange raft does this help?

[strange raft]

Uh..so i've just started learning this can you like put that in words

[ionic jewel]

oh you arent doing summation work

[ocean seal]

Chai T. Rex

[ionic jewel]

i assume chai is typing a good answer

[oak chasm]

If 1 doesn't work, try 2.

Depending on whether the natural numbers are considered to contain 0 or not.

[ionic jewel]

the way i was answering it you sum up all the numbers from 1 to n, then divide by n

and if you happen to know what the sum of numbers 1 to n is, then its easy to do that way

otherwise using chai's intuition way would be better

[alpine sable]

what’s 14 to the power of 8

[ocean seal]

Chai T. Rex

[oak chasm]

@alpine sable Use a calculator.

[alpine sable]

oh ok thanks

[ionic jewel]

@oak chasm 0 is not in natural numbers in this problem

wouldnt make sense

almost certainly 1 to n

[strange raft]

yeah that's what i thought

[ionic jewel]

does 0 to n give an answer?

no

its 1 to n

[oak chasm]

Sure it does.

[ionic jewel]

not one of the choices

[oak chasm]

Look at the linked work.

[upbeat wagon]

WHTA 1+1 IM SO CONFUSED

[oak chasm]

It matches one of the choices.

[strange raft]

So it will certainly be n+1/2

[ionic jewel]

oh yeah my bad

im pretty sure its 1 to n still

[upbeat wagon]

hi

[alpine sable]

[ionic jewel]

if you used parenthesis, this is correct for 1 to n

[upbeat wagon]

ummmfmfmfmffmfmfmfmfm do u guys do area model?

[strange raft]

since if i take 1 in the formula the mean will be 3/2 thus n+1/2

[upbeat wagon]

[ionic jewel]

yes

[strange raft]

parenthesis why?

[oak chasm]

n + 1/2 uses PEMDAS.

The division is done before the addition.

(n + 1)/2 does the addition before the division.

[strange raft]

oh my bad

thanks for helping tho

[oak chasm]

No problem.

[uneven stratus]

Is this free?

Does anybody know a way to show this without L'Hopital's rule? I have tried several different things, but none have worked.

[shell widget]

maybe expand out (x+1)^n?

[uneven stratus]

O, that is one thing I have not tried xd thought it would be too much toil

But do you think that could work?

I will try it, then

[shell widget]

no its actually not that difficult

and yes that will work

[next pulsar]

binomial theorem is a beautiful thing

[glass lichen]

it's first principles

[next pulsar]

but yes moshi is correct

[glass lichen]

$\lim_{h\to 0}\frac{(1+h)^n-1^n}{h}$

[ocean seal]

moshill1

[glass lichen]

to make it more obvious

[uneven stratus]

OH, cannot believe I did not see it; it is a derivative

[glass lichen]

yes, f(x)=x^n, and you want f'(1)

so n(1)^(n-1)=n

[uneven stratus]

Haha, that is true xd thanks
Now I see why expanding might work (as I cannot use that fact to solve this)

[glass lichen]

yeah, expanding it out would basically just be proving power rule again

[uneven stratus]

Yea xd

[next pulsar]

How can I quickly show that the left polynomial is not a factor of the other? I really don’t want to do long division...

[slender marten]

I suppose you could find the solutions of the quadratic and if the quadratic is meant to be a root of the polynomial P then P(x_1) = P(x_2) = 0.

[lost steeple]

https://gyazo.com/4b6e56f79270d729660fce6f1b4566f5

How does one do this? Can someone explain to me the steps

[glass lichen]

3 or 4?

[lost steeple]

Both

Like how can I find the value of (x)

and the value of (y)

for 4.

actually ignore 3 I think I'm capable of that

[glass lichen]

ok well if sin(x) is 5/13, and x is in quadrant 2, you can draw a right triangle in that quadrant with signed side lengths

namely an opposite side of 5, and hypotenuse 13, so the missing side will be on the negative x-axis and have a value of -12

so cos(x)=-12/13

[next void]

I am trying to solve this question but I got part b incorrect. Can someone take a look and explain where I went wrong?

There are about 2 million births per year in a certain country and about 4400 hospitals. For the sake of simplicity, suppose the births over a ten-year period were evenly distributed over all the hospitals, and suppose each of those hospitals divided their births into strings of 19 consecutive births. Assume that boy and girl births are equally likely.

a) What is the probability of 19 births in a row being boys?
b) What is the probability, in any one of those hospitals, of having a string of all boys?

[lost steeple]

Ok thanks I'll give it a try

[red sail]

anyone know what to do here

[noble sinew]

Just round your answer

and you get the same

[blazing rose]

no idea what to do here

[glass lichen]

solve for k using the 24g/day

[blazing rose]

so ik that 24 is a rate

oh wait

24 = ky

?

[glass lichen]

24=k(1)

since every day it increases 24g/day

[blazing rose]

why is y = 1

[next void]

@noble sinew How would I round it? If I type in 4.5768x10^-4 into my calculator then I get 4.5768E-4 and not 0.0005 which is the correct answer. Can you explain it to me.

[glass lichen]

cause after 1 day, the rate of change is 24g/day

[noble sinew]

what does 4.67768*10^(-4) mean?

it means 0.00045678

[signal brook]

türk var mı varsa dm çok acil

[blazing rose]

i dont understand

can you go through it from step 1

[glass lichen]

Ok, solve for k

[blazing rose]

dy/dt = 24

so we set 24 = ky

[glass lichen]

yes

[blazing rose]

now where did u get y = 1

[glass lichen]

cause it's been 1 day

[blazing rose]

because it says at t = 0 dydt = 24

[glass lichen]

dy/dt = kt, every day it increases by 24g/day

[blazing rose]

howd u g et dy/dt = kt

its ky?

[glass lichen]

ky yes

whatever

[blazing rose]

oh ok

so its increasing at a rate of 24g/day ok

but that is true when t = 0

[glass lichen]

Ok let's do it differently.. have you solved the DE?

[blazing rose]

de?

what does that stand for

[glass lichen]

$\dv{y}{t}=ky\implies \frac{1}{y}\dd{y}=k\dd{t}\implies \ln(\abs{y})=kt+C\implies y(t)=Ae^{kt}$

[blazing rose]

differential equation

ah

[ocean seal]

moshill1

[blazing rose]

y = Ae^(kt)

[glass lichen]

yes

[blazing rose]

is that a formula i should know?

[lost steeple]

@glass lichen Hey so I found the 12 but how do I find "x"?

https://gyazo.com/8bfb880dc7543dd13c21e97d4d674bce

[blazing rose]

bruh

[glass lichen]

I mean it's the general solution to y'=ky

[lost steeple]

@blazing rose Srry to interrupt

[blazing rose]

oh got it

[glass lichen]

but just do the seperation, it's not hard

[blazing rose]

yeah ok ok

then u just plug in ?

[glass lichen]

anyway, t=0 means y=120

so A=120

[blazing rose]

(0, 120)

ok

[next void]

@noble sinew Thank you for the explanation. I should have realized that.

[blazing rose]

so A = y intercept?

[glass lichen]

so then $\dv{y}{t}=120ke^{kt}$

[ocean seal]

moshill1

[glass lichen]

and when t=0, dy/dt = 24

so k=24/120

[blazing rose]

ah

wait

120(24/120)e^(kt)

[glass lichen]

what about that?

[blazing rose]

simplifies to

24e^(kt)

?

[glass lichen]

yeah, and?

[blazing rose]

not an answer choice

[glass lichen]

cause you're not asked for dy/dt =

[blazing rose]

oh right

[glass lichen]

you're asked for y=

[blazing rose]

y(t)

yeah

[blazing rose]

ok ill try to work this out thanks alot

[flat pagoda]

?

[blazing rose]

12^ + 16^2 = x^2

and solve

144 + 256 = x^2

x^2 = 400, x = 20

[vale wigeon]

next time please don't just do people's problems for them like this

[blazing rose]

oh

sorry

[alpine yacht]

can someone tell me how to start this one?

i am confused

[glass lichen]

want a such that $\lim_{x\to a}\frac{x^2-a^2}{x-a}=15$

[ocean seal]

moshill1

[alpine yacht]

but, what do you plug in for x to get a value for a

@glass lichen

[glass lichen]

solve the limit and set it equal to 15

[alpine yacht]

can i just plug in any number for x?

i dont know what to do from here, i am stcuk

[glass lichen]

solve the limit, what do you get?

[alpine yacht]

wait, do you plug 15 for a?

[glass lichen]

no

solve the limit

[alpine yacht]

idk what to do with 2 varaibles

[glass lichen]

$\lim_{x\to a}\frac{x^2-a^2}{x-a}=\lim_{x\to a}(x+a)=2a$

[ocean seal]

moshill1

[alpine yacht]

what do i do after i find the limit

[glass lichen]

solve for a

Like I said already

[alpine yacht]

so, i set up 2a+15?

a = 15/2

[glass lichen]

yes

[alpine yacht]

ok thanks

[cyan token]

How many 3 digit numbers can be formed using only the digits from 1-6 if the number must contain the digit 5

<@&286206848099549185>

[topaz cloud]

hi

Solve the exercises by the rule of three.
NOTE: With calculus!

a) Calculate 20% of 84.
b) How much is 8% of 92?
c) If 578 corresponds to 100%, 200 corresponds to what percentage?
d) If 480 corresponds to 100%, 150 corresponds to what percentage?
e) 15% of 3000
f) 32% of 1500
g) 40% of 180
<@&286206848099549185>

[lone heart]

From

#❓how-to-get-help

Rule 4

If your question has not been answered for a minimum of 15 minutes, you may use the Helpers tag once. Please do not try to bump your question using this ping unnecessarily. Do not abuse this ping. Do not individually ping users with the Helpers tag without their express permission.

[final kiln]

Hello, I need some help with some cal 1 problems

[solemn ore]

can some1 help me

[final kiln]

Find the slope and y-intercept of the line through the point (9,9) that cuts off the least area from the first quadrant.

[slender marten]

What have you tried?

[final kiln]

here thats my question

well 1 of many

[slender marten]

Can we all go to different rooms?

Instead of 10 in one.

[solemn ore]

<@&286206848099549185>

[slender marten]

What is the obsession with the immediate helper pings?

[cyan token]

People are impatient

[slender marten]

Just makes me not want to help.

[cyan token]

Can you help me?

[slender marten]

I don't know how to do that question. 😢

[gray gorge]

Read #❓how-to-get-help

[raw anvil]

oof my bad

[wild shard]

[bright wind]

How can u

[sinful lintel]

Hi! Does anyone know whether the Lagrangian multipliers method can yield critical points that are saddle points?

^ multivariable calculus

I'm attempting to show that a solution I found by using the Lagrangian method is the maximum but the following reasoning:
(1) I found only one critical point, so that point must be the absolute maximum or the absolute minimum
(2) if I pick another point and its value is less than that of the critical point, then the critical point is the absolute maximum

But if the Lagrangian yields saddle points, then I can no longer use this reasoning

[slender marten]

Critical points can be saddle points.

You can use the second derivative test to determine if it's a local minimum or local maximum though.

[stark blade]

Any help with this guys? part a = 9x^2 + 3 but what is part b??

[slender marten]

The derivative of the denominator is 9x^2 + 3 = 3(3x^2 + 1). Therefore if you factor out a three then the top is the derivative of the bottom hence you can either know the answer is 1/3ln|bottom| + c or use substitution with u = 3x^3+3x to verify this is indeed true.

[lyric lynx]

You can also just plug the solution into the original function(s) and show that its a max

[lethal wolf]

smart

[alpine sable]

<@&286206848099549185> I need some guidance quite stuck

(Simultaneous Equations)

Idk what I did tbh

Threw some maths at it :))

[slender dirge]

The Apex Community Center has a 50 foot by 40 foot area outdoors in which to build a swimming pool. The pool will be surrounded by a concrete sidewalk of uniform width. What could be the width of the sidewalk be if organizers want the pool to be 1656 square feet?

[cyan token]

<@&286206848099549185> help

[shell widget]

factor out x so you have x^9 * [1 - (3/x^3)]^9 and then use binomial theorem which should be easy

[stark blade]

wouldnt it just be 80% of 1656 ? @slender dirge

since height is 80% of width and that '1656 sq feet' is the width

[amber shoal]

help

[stark blade]

a I believe

10,000x0.88^5

[slender dirge]

no the area is 1656

[stark blade]

ah then i think it's 920 @slender dirge

[amber shoal]

ty

[worthy verge]

Only question c i encountered a problem

[manic quail]

,rotate

[worthy verge]

Answers are x=1 y=-1 z=0.5

2c

i tried inversing every single equation for question C but ended up at a dead end

have to use cramer's rule, and some kind of matrix method to do it

<@&286206848099549185>

[slender marten]

There's a nice way to remember Cramer's rule.

[slender marten]

Stop abusing the tag...

The trick is this.

You can write it in the form Ax = b.

Where x = [u v w]^T (stand it up).

[glass lichen]

$\begin{bmatrix}u\v\w\end{bmatrix}$

[ocean seal]

moshill1

[slender marten]

To solve for u, the first one, you do the fraction of two determinants, the numerator being the determinant of the original matrix A but replace the first column, the way to remember this is the u is the first one in the vector, with the vector b, then the denominator is just the determinant of A.

Similarly for the v, the second one, you do the fraction but in the numerator this time you replace the second column with the vector b.

The exact same pattern for solving for w.

This is why to have a unique solution, the determinant of A must be non-zero, since Cramer's rule has it in the denominator and of course the classic divide by zero error.

[manic glade]

how can i prove/disprove this function is injective? f: Q->Q x<=0

[shell widget]

for one-to-one or injectivity, if we have f(x_1) = f(x_2), we must have x_1 = x_2

[worthy verge]

I understand how to use cramers rule i dont understand how to use it when the unknowns are the denominators

[manic glade]

but how do i go about doing that

[slender marten]

Oh, I was looking at (a).

That is quite the conundrum.

[worthy verge]

acutally, now that i think about it, i just ignore the fact theat they are fractions and continue like they were whole numbers?

[worthy verge]

Then i will get this but wiht 1/x and so on

so i just inverse both sides ?

If you get what i mean

[slender marten]

Maybe just write 1/x as x^(-1), 1/y = y^(-1) and 1/z = z^(-1)and let these be the things in the vector.

[worthy verge]

.

Ok give me a minute ill try it out

[slender marten]

Yeah, I think that will do it, then just solve for 1/x, 1/y and 1/z as normal.

No reason for it not to, could just say 1/x = a, 1/y = b and 1/z = c, I guess you just have to know any solution on a coordinate axes does no exist.

[worthy verge]

My answer turned wrong

;v

[slender marten]

I can try it.

Them compare answers.

[worthy verge]

i got x^-1=3, after finding determinant x diving by determinant made up of numbers on the left of equation only

if you know what i mean

[slender marten]

I got x = 4.

1/x = (-3)/(-12).

[worthy verge]

;v...

[slender marten]

I get answers.

I'll take a picture of my quick work.

x = 4, y = -4 and z = 2.

They don't seem to satisfy the first equation though.

I just recheck my main determinant in the denominator.

[worthy verge]

my det is -3

[slender marten]

Ok, that's my error.

No idea how I got -12, I double check what I entered.

[worthy verge]

thats a clever way to represent your answers i really like the presentation

ill copy

your style

right so my det of X is incorrect, ill redo everything agane

[pale terrace]

hi can someone help me get started with this question?

[bold mirage]

find the exact solutions

They defined sin for you

It should be easy-

[pale terrace]

I know it's an easy question but I just don't see how to start

do you have any ideas for part c at least?

[worthy verge]

yep i got the answers, thanks dude!

[slender marten]

Nice. 🙂

sin(2x) = 2sin(x)cos(x), you should be able to use your results of sin(x) and cos(x).

[bold mirage]

sin 2x = 2 * 4/5 * 3/5 = 24/25

Since sin x=4/5 we have a (3, 4, 5) triangle and we can define all sines and cosines, cos x = 3/5

Wait-

Oops

I forgot negative

But you get the point

[brave aurora]

could I get help with question 6?

this is the answer btw

I just dont know how they got it

[wispy urchin]

What is rationalizing a denominator and how do you do it?

[ionic jewel]

get rid of radicals in the bottom, usually by multiplying by the opposite thing

i forget what its called

$(a + \sqrt{b})(a-\sqrt{b}) = a^2 -b$

[jagged imp]

Conjugate

[ocean seal]

bunny

[ionic jewel]

you use this property

yes conjugate thank you

[wispy urchin]

Oh thank you

I hope exponential and logarithmic functions aren’t hard. I’m about to learn about whatever that is.

[ionic jewel]

logs take a bit to get used to but arent so bad

exponents are fine

both of them involve memorizing some rules

[wispy urchin]

Thanks, bunny. You must’ve been the student who everyone went to for help. lol

See ya wish me luck

[ionic jewel]

haha gl

[proven fossil]

Emily has 20 metres of fencing. She needs to form an enclosure in the garden for her dog. What is the maximum area she can enclose if all the fencing must be used?

My answer is 25, is this correct?

[oak chasm]

@proven fossil How did you get 25?

[proven fossil]

I found the perimeter, 5+5+5+5, then the area 5x5=25

[oak chasm]

OK, are there shapes other than squares where the fencing can surround a larger area?

[proven fossil]

no

[ionic jewel]

thats pretty shapist

[proven fossil]

I assume this question requires trial and error and you just have to find the maximum area in which she can use

[ionic jewel]

why only squares?

[oak chasm]

@proven fossil If you want to maximize area, there's a shape better than a square.

[proven fossil]

I used a rectangle

[ocean seal]

Chai T. Rex

[ionic jewel]

wait he already did that tho

doesnt really lead him on the direction to the actual answer

oh wait i might see nvm ignore me

[hardy geyser]

I need help with 2

It’s b cause there vertical? ?

also can someone help refresh me on why it’s two I haven’t don’t geometry in. A while

[slender dirge]

The Apex Community Center has a 50 foot by 40 foot area outdoors in which to build a swimming pool. The pool will be surrounded by a concrete sidewalk of uniform width. What could be the width of the sidewalk be if organizers want the pool to be 1656 square feet? Quadratic standard form, any help/

?

[glass lichen]

yes, a+c is not guaranteed to be 180

[sinful cloak]

i just need help with my test

[glass lichen]

no help with tests

that's called cheating

[slender marten]

^

[sinful cloak]

how old are you

[glass lichen]

19, dont see why it matters

[sinful cloak]

tryhard arent you

[glass lichen]

Eh, I just know how to study and not need to cheat

[sinful cloak]

sheeeeeesh

[slender marten]

Supplementary angles are two angles that add up to 180 degrees, a straight line. Why do Americans insist on naming everything... I had to look it up. Complementary angles are two angles which that dd up to 90 degrees, a right angle.

[slender dirge]

The Apex Community Center has a 50 foot by 40 foot area outdoors in which to build a swimming pool. The pool will be surrounded by a concrete sidewalk of uniform width. What could be the width of the sidewalk be if organizers want the pool to be 1656 square feet? Quadratic standard form, any help/

[cyan token]

Ah yeah just gonna keep this down here thanks!

[gray isle]

@slender dirge what have you tried?
have you drawna diagram

[slender dirge]

drawing, factoring etc

but im shit at factoring

[gray isle]

what do you have atm

[slender dirge]

well, i got to an equation, but i dont think its right and not even close

[gray isle]

can you show your diagram and what you did

[slender dirge]

4x^2+180x+344 after factoring

lemme take a picture

sorry for bad handwriting

[ashen wave]

I live in Apex :> plus im learning the same stuff u are

[slender dirge]

what school lmao

[gray isle]

where's the rest of the work

[ashen wave]

Apex Friendship middle

[hardy geyser]

@glass lichen so b and d is not guaranteed 180

[slender dirge]

YOOOO SAME

no way

[ashen wave]

NO U DONT

[slender dirge]

yes

[ashen wave]

what grade are u in

[slender dirge]

8th mcconnell homeroom

[ashen wave]

LMAO

LOL NOOOO

I was in her class but I switched to virtual!

[slender dirge]

damn

[ashen wave]

I was there last week

whats ur name?

[slender dirge]

nick

[ashen wave]

where u in the back?

[slender dirge]

right side towards the back

[ashen wave]

lol your smart for using discord

[slender dirge]

if your looking towards the tv

lol

[alpine sable]

how solve

[ashen wave]

I was placed in the front rip

[slender dirge]

rough

she doesn't care about me cause i have a 99

whats your name?

[ashen wave]

yeah science is too easy

Alondra

[slender dirge]

last name

[ashen wave]

Lukasiewicz xD

[slender dirge]

ahh

i know a different alondra

[ashen wave]

what? I thought I was the only one ;-;

[slender dirge]

at least she was at our school last year

idk about this year

[ashen wave]

Whats ur last name?

[slender dirge]

sayre

[modern yarrow]

You and your friend get summer jobs grooming dogs! The following chart shows the number of dogs groomed over your first week of work.

a) Determine the average dogs you groomed over the first week.

b) Determine the average number of dogs your friend groomed over the first week

c) Determine the standard deviation for the sample of dogs you groomed over the first week

this is dumb actually

nvm

[slender dirge]

1. just add all the dogs you groomed and divide by 7 2. do the same for your friend

[modern yarrow]

oh ok thanks

I have another question

when they're talking about who is more consistent

who has less variation right?

so whichever standard deviation is smaller?

[slender dirge]

yes

[modern yarrow]

ty

[slender dirge]

np

[teal patio]

uhhh why the gif? lol

[hardy geyser]

Wrong chat

[modern yarrow]

At a swim meet the fastest swimmers will advance to the provincial competition. The swim times for the 200m freestyle are normally distributed with an average time of 2 minutes and a standard deviation of 6.7 seconds. Only the top 6% of swimmers can advance. Whatt is the cut-off time to qualify for provincials?

I'm having a hard time with this

[alpine sable]

Geometric series

[ionic jewel]

I'm not sure if i know the math anyways but i can't even read it

ah there u go i guess

[teal patio]

so botnuke from left side of = to the right side is a geometric series?

on top line

[alpine sable]

No, that top line looks like just rewriting the exponents

[teal patio]

maybe a log rule?

[alpine sable]

a^(bc) = (a^b)^c

[teal patio]

ooo ok

botnuke do u mind explaining to me what a geometric series is?

[alpine sable]

About to go to bed but you can try reading this

https://tutorial.math.lamar.edu/classes/calcii/series_special.aspx

[teal patio]

ok ty

[buoyant kayak]

in simple terms, its the sum of an infinite sequence of numbers with a common ratio

[alpine sable]

It’s a standard topic so there should be plenty of other resources on the internet about it if that’s not suitable for whatever reason

[teal patio]

hmmm ok

[stoic verge]

I need to use the integral test for 1. But I do not know how to integrate that

Can someone help me

[modern yarrow]

At a swim meet the fastest swimmers will advance to the provincial competition. The swim times for the 200m freestyle are normally distributed with an average time of 2 minutes and a standard deviation of 6.7 seconds. Only the top 6% of swimmers can advance. Whatt is the cut-off time to qualify for provincials?
I'm having a hard time with this

[haughty dragon]

are you sure? looks like a comparison test might be better for this question

[hollow kelp]

What about ratio?

is that a possibility?

Actually no I think comparison is better

[normal shuttle]

how do i find the length for this?

[stoic verge]

the given function is smaller than 1/n and 1/n diverges. so that doesn't really tell me anything, right? @haughty dragon

[modern yarrow]

can someone please help me with a probability question

[haughty dragon]

mmh youre right

[haughty dragon]

alright there might be a better solution, but 1/(7+n) diverges and it's smaller than the series you're given

[stoic verge]

That's true thanks @haughty dragon

[left ginkgo]

Can someone help me with this problem :<

[stoic verge]

@haughty dragon my prof replied and if youre interested, he told me to integrate 1/nlogn and then do limit comparison with that instead

[haughty dragon]

ohhh interesting

i did try limit comparison but didn't think of that

[hardy geyser]

Help

I said D

[slender marten]

a + b = 180 and a + c = 90.
(a + b)-(a + c) = 90. a + b - c = 90 degrees, it seems you are right.

[dawn wraith]

Do you still need help with that?

[alpine sable]

i thought it was b-c=90

can somebody help?

[sterile zealot]

I can

75% of the figure is 162

With this measure, can you determine how much is NOT included in the 75%?

[normal shuttle]

i already solved

than you doe

[tacit breach]

<@&286206848099549185>

[sterile zealot]

Alright, @tacit breach

[tacit breach]

i think i can split this into 2 triangles and a rectangle

[sterile zealot]

Yes

[tacit breach]

and do 26-16/2to get both sides and easily get the area for those 2

[sterile zealot]

Well dang, you've got it

[tacit breach]

Almost

I need to get the height for the rectangle

[sterile zealot]

Pythagorean theorem

[tacit breach]

ah i see

Did you get 322

[sterile zealot]

...is that 21 * 12?

[tacit breach]

I think I did it wrong

No its 16*12+130

the area is 65 for both triangles,

[sterile zealot]

Ah-ah!

[tacit breach]

actually its not

[sterile zealot]

The diagonal is not the height

[tacit breach]

yeaah 🤦 didnt notice i did that

[sterile zealot]

Common mistake, my friend

Even then, each triangle is half

[tacit breach]

oh well

[sterile zealot]

Putting them together would be the whole value of the triangles's length times the height

[tacit breach]

Yeah

[sterile zealot]

Combine that with the rectangle...

Alternatively, once you find the height you can just do (16 + 5) × h

Hence my asking if that was

21 × 12

[tacit breach]

Yeah lol

[sand notch]

can someone help me on part a

[vale wigeon]

@sand notch do you still need help w/ this

[kindred helm]

right triangle, hypotenuse 10 other lines are 6 and 8

i have to calculate the centroids distance from the hypotenuse

how to do this?

[vale wigeon]

are you ok with coordinate-bashing

[kindred helm]

the tip provided in my textbook says: i gotta calculate the centroids distance from the other lines first

then somehow use the area of the triangle to my advatnage

i have 0 clue i have never seen something like this

[vale wigeon]

can you make a diagram?

maybe things will make a little more sense when you can see what's going on.

[kindred helm]

u mean a picture?

sketch it out?

[vale wigeon]

sure, if you want to call it that

[kindred helm]

ive done that on paper

[alpine sable]

if you find the centroid you'll see something interesting

i won't spoil it but you don't need to coordinate bash

[kindred helm]

im assuming all the medians still work in right triangles

as they do in normal triangles

[alpine sable]

of course

[vale wigeon]

why would they not

[alpine sable]

draw them

[vale wigeon]

also show us your picture

[kindred helm]

ill do on geogebra

[cosmic bison]

I need answers quick lol

[vale wigeon]

@cosmic bison no

we do not give out answers here

[cosmic bison]

rip well can you help me on the chart ones

I don’t understand

[vale wigeon]

also you are intruding in an occupied channel

read #❓how-to-get-help

[kindred helm]

okay well im assuming that making a height line from the right angle towards the hypotenuse should help me

that goes through the centroid

[vale wigeon]

the altitude won't go through the centroid

[kindred helm]

ah okay

[vale wigeon]

...not necessarily anyway

[kindred helm]

huh

[vale wigeon]

okay yeah good

[kindred helm]

so

ah

AFB

[vale wigeon]

give a name to the centroid

say M

you can actually drop perpendiculars from A and M onto BC and establish a pair of similar triangles

since the medians in a triangle split each other in a 2:1 ratio

[kindred helm]

doesnt AFB have 2 same lines

[vale wigeon]

?

[kindred helm]

the triangle AFB

FB is 5

AF is 5

AB is 6

[vale wigeon]

oh, AFB is isosceles

i overlooked that because my suggestion had nothing to do with it

[kindred helm]

so the distance from the centroid to the hypotenuse should be 1/3 * 5?

hmm its not

i checked the answer

[vale wigeon]

i didn't say it should be 5/3...

[kindred helm]

yeah i did

it has to be 5/3

my textbook has it as 8/3

no way its that big..

[vale wigeon]

?!

we are finding the distance to the hypotenuse are we not

[kindred helm]

yea

[vale wigeon]

that would not be 8/3

[kindred helm]

yes

[vale wigeon]

if anything, assuming i didn't fuck up, it's 4/5.

[kindred helm]

well my textbook has it as 8/3 and i think its 5/3 because the medians cut in 2:1

bcs AF is 5 and after the line goes through the centroid

its 1/3

[vale wigeon]

AF is not perpendicular to BC tho

[gray isle]

,w distance from (2,8/3) to x/6+y/8 =1

[ocean seal]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by ramonov#0921

[kindred helm]

wtf im blind it is 8/5 in my textbook

why did i look it as 8/3

i still dont get it lol

maybe i need to think about it for a bit longer

[gray isle]

well you should have the coordinate of the centroid and equation of the hypotenuse

[kindred helm]

the tip in my book was to use the area of the triangle and the distance to the other lines, you got an idea how this is done?

distance from the centroid

[gray isle]

1 sec

[kindred helm]

if u know can say a few hintss

i wanna try at least

[gray isle]

ok. drawing lines from the centroid to the vertices gives you 3 triangles with the same area

[knotty garnet]

even better

[kindred helm]

oh true it does

[knotty garnet]

if you draw all 3 medians, you get 6 triangles all of equal area

after you prove that

i recommend proving the following statement:

The centroid divides each median into two segments with length ratio 2:1.

these are two extremely useful concepts regarding centroids that anyone working with them should know

[gray isle]

don't really even need that applying the previous property

[knotty garnet]

yeah i just think it's good moving forwards

[kindred helm]

okay now i know that theres 6 triangles with area 4

and the big triangle is 24

i just solved it

wait no i didnt

yeah cnt solve it

trying to solve it how my textbook tells me to

[gray isle]

the perpendicular (shortest) distance from the centroid to the hypotenuse will be the altitude of your triangle

[kindred helm]

yea the height

or does altitude mean a diff thing here

[gray isle]

more or less the same

consider the triangle BCM will have an area of 8

[kindred helm]

yep

[gray isle]

treat the hyp of the big triangle as the base (which has length 10)

and then apply formula for area

1/2 * 10 * h = 8

[kindred helm]

how could i miss that

i tried making it way too complex

ty

[wanton spoke]

it's a little blurred, but my answers are

1. OE bisects <HOP - Given
2. OE perpendicular HP - Given
3. OE = OE - Reflexive Property of Congruence
4. HE = PE - definition of a midpoint

this is all i can think of...

[ionic jewel]

doesn't bisect mean two equal parts, and then from what you have + that you have two sides and two angles of both triangles, so you could get the rest to prove

[wanton spoke]

oh yeah. I didn't add it because there's no other right angle sign in the right side and I thought they're not equal because of that

[ionic jewel]

it has to be 90 degrees

[alpine sable]

bruh i have no idea what this question means can anyone help

<@&286206848099549185>

[lone heart]

From

#❓how-to-get-help

Rule 4

If your question has not been answered for a minimum of 15 minutes, you may use the Helpers tag once. Please do not try to bump your question using this ping unnecessarily. Do not abuse this ping. Do not individually ping users with the Helpers tag without their express permission.

[alpine sable]

whoops

[alpine sable]

pls ping me if you know the answer to this

find the probability that a number between 1 and 10^n is a perfect square in terms of 'n'

can't seem to find the sol without using the int() function

posted here @keen widget''welcome :)

[alpine sable]

that is not me LOL

oh god

im sorry other tejas :(

[alpine sable]

( # of perfect squares in the interval [1,10^n] ) / (10^n)

floor ( sqrt(10^n) ) / 10^n

[alpine sable]

oh wait you wanted a formula that doesn't use the floor function?

i got something similar but im trying to find a sol without using functions like int(), floor(), etc.

ye 😂

thx tho

[teal rune]

uh

anyone know

really basic trig

that can help

my mind isnt working rn

[alpine sable]

sure

[teal rune]

oh cool

#22

I tried doing

using cosine

because i labelled the opposite as 45 and 30 on adjacent

[alpine sable]

angle of elevation is 39 deg

[teal rune]

yes

[alpine sable]

so its a 90-39-51 triangle

use the law of sines

[teal rune]

agreed

[alpine sable]

so sin 39/ 45 = 1/x and you can substitute

if it isn't clear i can write the sol and send a pic in 10 mins

[teal rune]

but 39 isnt a side?

[alpine sable]

no 39 deg is the angle of elevation

given

[teal rune]

yes

[alpine sable]

shall i?

[teal rune]

sure

it helps

i thought of soh

opposite/hyp

[alpine sable]

alr i just need some time 😁

[teal rune]

yeah my end formula was

x = 45/cos39degrees

[alpine sable]

i think you confused cos with sin

or wair

there's another way

[teal rune]

i just did sin

and it didnt work

ended with

x = 45/sin39

can u send a pic of the triangle u made

@alpine sable

[alpine sable]

one moment discussing a problem with a friend