help-23

[cobalt plinth]

I’ve tried grouping but it doesn’t work, I factored out a 2 which gets me 3x^3-9x+6x-10 and since I’m finding the x int the 2 can just be divided out

But idk what to do next, I’m stuck

[inland ivy]

Maybe your teacher wants you to use that rational root theorem thing

You know where you divide all of the possible roots of the first and last terms and plus minus it

Then find the rational ones and trial and error them

So the roots of a are

1,2,3,6

The roots of d are

1,2,4,5,10,20

So then you have

roots of a / roots of b

±1, ±2, ±3, ±6, ±1/2, ±2/5 ...

Once you find one by plugging in the root into the original equation you can just long division the thing by that root

And factor it normally as a quadratic

Just dont forget your trial and error factor that you divided out in your final answer

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[cobalt plinth]

LOL tbh I haven’t heard of that theorem yet but ty

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[frosty ridge]

I hypothesized that the sequence will converge to 0. My thinking was that sin(whatever) will bounce between -1 and 1, while the denominator will become arbitrarily large.

I want to find a value for N such that a_n-a is within the tolerance. so I have the inequality |a_n-0|<epsilon. I'm trying to isolate for n, but I'm stuck. Anything guidance to start off the problem would be appreciated.

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@frosty ridge Has your question been resolved?

[quasi bison]

@frosty ridge consider that |sin(whatever)| ≤ 1

and so |a_n - 0| ≤ 1/|n^2 - 3|

that ought to make your life a lot easier

[frosty ridge]

thanks Ann. Ill try this.

.close

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[lean otter]

I was following a solved example but I can not figure out what happened in the 2nd step and onward

[indigo veldt]

What they do is multiply through by 2^x (a positive value), bring everything to the order side, and then factorise it

You're basically factorising a quadratic (imagine the 2^x is a value like u, i.e. u^2 - 6u + 8)

[lean otter]

Oh

Makes sense now, thanks

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[umbral meteor]

Hi, i need to calculate the coordinates of the intersection between this plane and the x, y and z-axis

[umbral meteor]

.close

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[proper crypt]

The distance between A and B is 10.5 kilometers.
From A, person 1 walked to B.
After half an hour, person 2 also starts walking from A with the speed of 4 kilometers per hour.
Person 2 caught up to person 1, and immediately went back to A with the same speed.
Person 2 got to A and person 1 got to B at the same time.
Find the speed of person 1.

[proper crypt]

I've tried filling out a table with the information but I've already redid it 2 times and I am not getting to the answer.

Something like this:

and this is easy to fill out:

but from here I've tried a few things and none of them worked

<@&286206848099549185>

[marsh walrus]

hmm i like this problem

idk how exactly to solve it off the top of my head

my guess would be to start backwards

[proper crypt]

well do u have any idea how to continue filling out the table

[marsh walrus]

sorry if you have a system you use in your class

i dont really understand the table

[proper crypt]

p1 is person one

S is distance

T is time

V is speed (velocity)

if u have 2 of the 3, u can find the one thats left

[marsh walrus]

ah okay

[proper crypt]

and I need to somehow form an equation that involves x to find it

[marsh walrus]

well i gues syou know t right

[proper crypt]

maybe even add another variable and get 2 equations, no idea

[marsh walrus]

well you know a t somewhere

since its half an hour when they meet

[proper crypt]

so?

[marsh walrus]

maybe im not understanding the table what are the discrete moments you are filling in information

just when they leave and when they arrive in the end?

or at each point you have information

[proper crypt]

like

before meeting

is the journey from A to the meeting spot

and after meeting

for person 1 it would be from the meeting spot to B

and for person 2 it would be from the meeting spot back to A

[marsh walrus]

okay

[proper crypt]

ok maybe this will work?

marking the distance from A to the meeting spot as y

[marsh walrus]

so youre saying up until the meeting, theyll have both travelled the same distance

after the meeting, p1 will travel whatever he had left to travel

and p2 will just travel that same distance backwards

this gives you access to time in a convoluted way

🤔

well you have to go through y

and v2

after starting p2 waits for half an hour

then travels y distance at 4km/h

[proper crypt]

I think I'll find the time and build an equation out of it

but sec, im drawing the table on paper rn

[marsh walrus]

i was thinking of starting a completely different way but i have no idea if its fruitful or not

[proper crypt]

since the other one had eraser markings

[marsh walrus]

probably better to use a known method

I believe that since velocity * time = distance, you can use 1/velocity * distance = time

[proper crypt]

yeah

thats what I was gonna do

[marsh walrus]

oh, nice

[proper crypt]

ok so

now for the equations

what should I do

[marsh walrus]

im not sure ive never seen this method before

[proper crypt]

im trying something out

I think filling that as Y was a good idea

so

before meeting:

p2 time + 0.5 = p1 time

and

after meeting:

p1 time = p2 time

[marsh walrus]

how do you use the table

[proper crypt]

u just

how do I explain this..

u fill things out

and from the information u build equations

so right now

i'm using the time I found by distance/velocity to build the equations

ok I solved it... XD

[marsh walrus]

sorry i didnt help much

lol

[proper crypt]

when I ask stuff here I just become pro

[marsh walrus]

whats the answer

[proper crypt]

3

[marsh walrus]

bro me too dwai

[proper crypt]

I already knew its 3 tho

should I send u my work

if ur curious

[marsh walrus]

nah its alright

i wanna try it out myself

[proper crypt]

alright

[marsh walrus]

from the method i was thinking

[proper crypt]

x = 3

[marsh walrus]

i wanted to try setting speed 1 as a ratio of speed 2

and working from there

anyways

[proper crypt]

so I type .close now?

[marsh walrus]

yup 😄

[proper crypt]

alr thanks!

.close

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[hybrid sable]

How did this happen?

[torpid oxide]

divide nom/denom by cos(x/2)

[hybrid sable]

Ohhh I thought it used some trig identities that I didn't know lol

Thanks

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[waxen sphinx]

hello, I'm getting started to trigonometry, and I have a question, they told me that cos(x) = a/h, but how can I calculate it without knowing h and a?

[waxen sphinx]

what's the formula

[pulsar condor]

wdym

cos(x) is in fact adjacent/hypotenuse for right triangles...

and angle x of course

[waxen sphinx]

I mean this

how can you know that cos(30) is 0.86602540378443864676372317075294, without knowing A and H

[pulsar condor]

cos(30)=sqrt(3)/2

you draw the 30-60-90 triangle

[waxen sphinx]

oh

[pulsar condor]

knowing why cos(something) = something without triangles requires stuff much more advanced then what you get in day 1 trig

[waxen sphinx]

I don't really care, about getting into hard stuff

can you tell me how is it called?

I'll read about it, and learn it for myself

[pulsar condor]

$\cos(x)=\sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!}$

[flat frigate]

Mosh

[pulsar condor]

in the more general case of what cos(value) is

[waxen sphinx]

I see

tysm

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[obtuse plover]

@pulsar condor how do u know how to do things like this, like the formatting of it

[pulsar condor]

practice..

[obtuse plover]

but how do i do it if i dont even know how to

[pulsar condor]

Just learn LaTeX

[obtuse plover]

o

is there a guide for it somewhere

or like a yt video

[worthy hemlock]

Google

[obtuse plover]

thx

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[fallen nacelle]

I do not understand where I messed up or how my answer is wrong

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[humble venture]

wolfram says that it's right, so it might just be a thing on the website

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[errant ore]

where is my q

[velvet nebula]

.close

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[lean otter]

how would i factor this:

[lean otter]

Which number is larger: 2^300 or 3^200?

[next igloo]

(2^3)^100=8^100 and (3^2)^100=9^100, so 2^300<3^200

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[lean otter]

oh!

im so dumb

lol

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[wintry panther]

?

[thin bridge]

what's e

[wintry panther]

dunno

I know that 100+11 and 100-11

they are 111 and 89

but dunno how that helps

[thin bridge]

you have a "difference of two squares"

[wintry panther]

yes

and have to do it in head

[thin bridge]

do you know the factorisation for the "difference of two squares'

[wintry panther]

don`t know if I heard that before

[thin bridge]

look it up

oh multiposted

.close

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[lean otter]

what’s the j-function

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[gentle nest]

how do i write a sum for this sequence
s1=6, s2=15, s3=28, s4=45, 25=66...
difference between each term gets larger by 4 every time
cuz i have to find the first 20 terms
sum of first 20 terms*

[lean otter]

((n-1)+2)(n+2+(n-1)) works i think but please do not use that because it’s probably wrong

so maybe like sum from 1 to 20 of that equation

[gentle nest]

o

thx

[steel stag]

nah the formula is a quadratic because you get 4 after taking the differences twice

here's the formula for sum of squares

and for sum of integers it's just n(n+1)/2

[gentle nest]

hm

i got 835

wait

oops wrong thing

[steel stag]

so like the formula for s is ax^2+bx+c, you gotta find the coefficients

the usual formula x^2 has twice differences of 2, so your sequence should be 2x^2 + something

[gentle nest]

i got 2n^2+3n+1

[steel stag]

looks good

[gentle nest]

do i just plug in the formula for the quadratic and the n?

and 20 for the constant 1

[steel stag]

yeah you just split it up like that

[mental minnow]

do you see how it's the second difference which is 4, not first difference

[mental minnow]

and i would indicate with u1, u2, u3, u4 exc. instead of s1,s2,s3,s4 as it might confuse you

[gentle nest]

o ok

didnt know what a second difference was

thx

[mental minnow]

yeah if the first difference is constant then we are talking about an arithmetic sequence while if it's the second difference which is constant then we are talking about a quadratic sequence

this will help you mate.

[gentle nest]

yeah i only learned the basics of summations

[mental minnow]

so if 2a = 2nd difference you already know that a is 2 right? (4/2=2)

[gentle nest]

wym

so like

[mental minnow]

nth term of quadratic sequence is of the form: an^2+bn+c

right?

[gentle nest]

yeah

[mental minnow]

if you know that 2a= 2nd difference which you have discovered that is 4 then divide both sides by two and you'll find that a=2

2a=4

divide both sides by 2

[gentle nest]

oh

[mental minnow]

and you get a=2

[gentle nest]

ic

i was confused where u got that 4 from

[mental minnow]

Now you know that second term is 15 and first term is 6. Difference between two terms is 9 right?

you know that 3a+b= 2nd term - 1st term

3a+b=9

we know that a is 2 so we just plug it in the equation

6+b=9

now solve for b and you will discover it is 3

1st term is a + b + c. The first term is 6 and theferore 6=2+3+c

solve for c and you will discover that c=1

so your nth term formula will be: 2n^2+3n+1

Do you get it mate? @gentle nest

now work from that nth term formula to find sum of first 20 terms

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[lean otter]

Can someone help with 1c pls

This is what I've done so far

But I'm getting a negative angle?

[worthy hemlock]

You have to wait at least 15 minutes before pinging helpers

[lean otter]

Ohh mb

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[hollow vigil]

help

[tame raft]

(2k-2)^2 + (2k)^2 + (2k+2)^2 = 1208

[hollow vigil]

why is it 2k

cant it be 1k

[fading bridge]

2k since its even

[worthy hemlock]

Definition of even number is 2k

[fading bridge]

any number* even results in even

[hollow vigil]

oh

[fading bridge]

yep so u expand and calculate

4kk-8k+4+4kk+4kk+8k+4=1208

so 12kk+8=1208

12kk=1200

k^2=100

k=10 or negative 10

since it says positive

k=10

and so your sequence is 8 10 12

[hollow vigil]

ohk

thanks

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[viral current]

can someone pls explain how to solve this related rates problem step by step

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[raw summit]

Help pls

[pulsar condor]

what have you tried...?

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@raw summit Has your question been resolved?

[raw summit]

@pulsar condor I’m stuck on this

[pulsar condor]

ok... you've done everything correct so far.

[raw summit]

Is the final answer y=5/9x?

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[barren carbon]

help please

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[viral current]

w/ respect of time

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@barren carbon Has your question been resolved?

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[lean otter]

do you know what a tangent is

[distant steppe]

No

[lean otter]

it is a straight line that touches a circle at one point

does not have to be a circle but in this case it is

so which one of the lines on the image touch the circle at just one point

[distant steppe]

does the line have to be in the circle?

[lean otter]

no

[distant steppe]

Q ?

[lean otter]

that is the point

you need the line

[distant steppe]

PR?

[lean otter]

yes

[distant steppe]

O

So the answer is PR?

Cause PR is the line with the Q touching the circle at just one point

Could u possibly help me with this?\

[lean otter]

ok so the thing with tangents is

at the point they touch

you can make a line that will for a right angle

so in that pic, the line OT will be perpendicular to ST

[distant steppe]

Okay

[lean otter]

then you can use pythagoras

[distant steppe]

What would I do?

[lean otter]

do you know pythagoras' theroem?

aka pythagorean theorem

[distant steppe]

a^2+b^2=c^2

[lean otter]

yes

and you know c^2 = 46

the line SO

OT is just a radius

so OT = 22

so you have the two sides, solve for n

[distant steppe]

uh

So line SO is 46 and OT is 22

and then

[lean otter]

yeah

[distant steppe]

what do I do with 46 and 22

c^2 = 46 and b^2 = 22?

[lean otter]

yes

so solve for a

[distant steppe]

ohh

@lean otter

is it 40.4

[lean otter]

yeah

[distant steppe]

How abt this one?

[lean otter]

apply the same method

same?

[distant steppe]

Yeah I did the same method and it = 97.86

unless I did it wrong

Yeah I think I did it wrong

[lean otter]

you did it wrong

what is your length of OB

[distant steppe]

uhhhhhh 63...?

[lean otter]

yeah it is

OB is the hypotenuse

[distant steppe]

ohhh

[lean otter]

try again

[distant steppe]

57\

57.8

[lean otter]

yes

[distant steppe]

Is this one correct

I did the same method

[lean otter]

yep

[distant steppe]

Ok how about this one?

[lean otter]

what should the sum of all angles in a triangle equal to

[distant steppe]

90?

[lean otter]

no

[distant steppe]

ohh the sum

270??

[lean otter]

nope

180

[distant steppe]

ah

[lean otter]

the three angles in a triangle must add up to 180

[distant steppe]

yes

[lean otter]

so you see one of them is 41

[distant steppe]

yh

[lean otter]

and since there is a tangent line PQ perpendicular to line OQ

that is 90

because perpendicular lines are 90 degrees

so you have one unknown angle

you can work it out now

[distant steppe]

OHHHHHHHHHHHH

Okay so

since its 90

and we already have 41

90 + 41 =
131

180 - 131 = 49

so the answer is 49

[lean otter]

yes

[distant steppe]

Thank you dude

This is the last one

[lean otter]

use same method as some of the quesitons earlier

pythagoras

OP is the hypotenuse

c^2

[distant steppe]

hm

uhhhhhhhhhhhhhhhhh

[lean otter]

what is the length of OP

in terms of t

[distant steppe]

I donrt know

[lean otter]

we're not supposed to directly give the answer...

Oh Sorry didnt know

[lean otter]

do you see the part of the line that isnt labelled on OP

[distant steppe]

yes

[lean otter]

that's a radius

you know what else is a radius

OQ

so you can say that the length of that unlabelled part of OP is 34

[distant steppe]

Oh

Okay

[lean otter]

any line from the center to the circumference of the same circle is the same length

[distant steppe]

ok

55.7?

thank you for the help

.close

[safe radish]

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[untold walrus]

I have the polynomial (1/2)x^(-5/2) + (1/2)x^(-1/2) -3x^(5/2)

[untold walrus]

Does it simplify down to (1/sqrtx) - 2(1/2) ?

[distant fjord]

there cannot be a sqrt(x) in the denominator

[untold walrus]

Oh

Do I leave it at x^(-1/2) or try to use the conjugate?

[safe radish]

@untold walrus Has your question been resolved?

[untold walrus]

<@&286206848099549185>

.close

[safe radish]

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[spice torrent]

Question about chain rule.

[spice torrent]

How is this result derived? I'm especially confused about the delta12

<@&286206848099549185>

[sand kestrel]

leibniz rule, you will see that for every higher order, the coefficients will be the same as a power expansion

but in this case, this 1 and 2 refers to x and y

both being t

so first you have the second order partial derivative with respect to x

on the middle, mixed derivative, the order might not interfere

and finally the second order ... with respect to y

you multiply everything by derivative of x wrt t and derivative of y wrt t, which is equal to 1

so nothing changes

[safe radish]

@spice torrent Has your question been resolved?

[spice torrent]

@sand kestrel How would you derive it using Leibniz notation then. I'm looking at it but I don't really get it. For example, for the first derivative, you're supposed to get a term for f, but I don't see where it will pop up from

This is where I'm at

[sand kestrel]

(g(t)=f(x,~y)), being (x(t)=t,~y(t)=t), right?

[flat frigate]

SubGui

[sand kestrel]

so differentiating one time with respect to t would lead us to $$g'(t)=\dfrac{\partial f}{\partial x}\cdot \underbrace{\dfrac{\partial x}{\partial t}}_1+\dfrac{\partial f}{\partial y}\cdot \underbrace{\dfrac{\partial y}{\partial t}}_1$$

[flat frigate]

SubGui

[sand kestrel]

in which (\dfrac{\partial f}{\partial x}=\partial_1 f) to short

[flat frigate]

SubGui

[spice torrent]

Ooo, I see, now I get it

[sand kestrel]

the thing is you get 2 in front of the mixed derivative because of the Clairaut-Schwarz theorem, it holds for most functions that (\partial_{12}f=\partial_{21}f)

[flat frigate]

SubGui

[spice torrent]

Thanks a lot for the help (:

.close

[safe radish]

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[azure wadi]

i need help

[safe radish]

@azure wadi Has your question been resolved?

[blazing fiber]

Boy, should we solve your complete test?

[azure wadi]

yes

[worthy hemlock]

Make an attempt first, then come back

We're not here to do your work for you

[safe radish]

@azure wadi Has your question been resolved?

[azure wadi]

i try and i need help

[pastel dove]

What did you try for the first question?

[safe radish]

@azure wadi Has your question been resolved?

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[royal kiln]

notice that

$\frac{?}{7y(x+19)} = \frac{?}{y(7x+133)}$

[flat frigate]

Disorganized

[royal kiln]

so then

$\frac{2}{7x+133} = \frac{?}{y(7x+133)}$

[flat frigate]

Disorganized

[royal kiln]

these statements would be identical if

(...do you see it?)

by inspection,

? = 2y

so that

$\frac{2}{7x+133} = \frac{2\cancel{y}}{\cancel{y}(7x+133)}$

[arctic portal]

Would I multiply by y

[flat frigate]

Disorganized

[royal kiln]

I just distributed y inside the parentheses

which is multiplication, yes

[arctic portal]

Ok that makes so much sense I thought it was more complex

[safe radish]

@arctic portal Has your question been resolved?

[safe radish]

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[mystic bear]

what did i do wrong

nvm i got it

[safe radish]

@mystic bear Has your question been resolved?

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[safe radish]

@lean otter Has your question been resolved?

[neat oyster]

What's the deal with the paint

Assuming you meant planted and not painted and that my brain is operational at this late hour, it's a simple system of equations.

[quasi bison]

@neat oyster please don't give out answers.

[safe radish]

@lean otter Has your question been resolved?

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[pearl shoal]

hello

[pearl shoal]

i have the integral of 1/cos x dx

i tried to multiply by sinx/sinx and u sub

but it doesn't work

i know that 1/cosx is secx

[safe radish]

@pearl shoal Has your question been resolved?

[lean otter]

have you tried looking online?

https://youtu.be/jjyPb-aVmlE

very good video about it

[pearl shoal]

ill take a look

[pearl shoal]

.close

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[fading cradle]

if a-b=5, ab=-6. Find the numerical value of a^2-b^2.

[pearl shoal]

start by finding a

and then b

then you calc a²-b²

[fading cradle]

but our lesson is factoring

[pearl shoal]

ok

a²-b² is what

oh wait

[fading cradle]

(a+b)(a-b)

[pearl shoal]

well i don't erally know your lesson

i would just subsitute a = b +5 and replace in the other equation

and then calculate a²-b²

[fading cradle]

if i square (a+b=5) i would get the value of a^2+b^2. not a^2-b^2.

[clever wing]

a coin is flipped 3 times. what is the theoretical probability of getting heads all three times.

[thin wraith]

@clever wing Stick to one question channel and be patient.

Also, use an available help channel instead of hijacking the occupied ones.

[safe radish]

@fading cradle Has your question been resolved?

[safe radish]

@fading cradle Has your question been resolved?

[velvet nebula]

Consider (a+b)^2-(a-b)^2 @fading cradle

[fading cradle]

how about the ac test?

can i use it?

ac=mn, b=m+n where m and n are factors?

[velvet nebula]

where do you get c from?

[fading cradle]

in the form ax^2+bx+c

[velvet nebula]

but what you have in this question is "a-b=5, ab=-6"

there's no quadratic anywhere

[fading cradle]

yeah

a^4-b^4

[velvet nebula]

where did you get a^4-b^4 from

What is (a+b)^2-(a-b)^2?

[fading cradle]

how do i get the (a+b) value?

if (a-b)=5 and ab=-6 what is a^4-b^4?

[velvet nebula]

Try expanding (a+b)^2-(a-b)^2

[fading cradle]

(a+b)^2 is a^2+2ab+b^2 -(a-b)^2 is a^2+2ab+b^2

[velvet nebula]

???

[fading cradle]

expand

[velvet nebula]

that seems soomewhat off

[fading cradle]

why?

[velvet nebula]

-(a-b)^2 is a^2+2ab+b^2 ???

[fading cradle]

-(a^2-2ab+b^2)

[velvet nebula]

yeah, what happens when you combine it together to form (a+b)^2-(a-b)^2 ?

[fading cradle]

i got 4ab

[velvet nebula]

how would it help to solve for a+b?

[fading cradle]

i dont know. its their difference?

i dont really know

[pulsar condor]

are you asking is there a difference b/w a+b and a-b..?

[fading cradle]

no.

[pulsar condor]

$(a+b)^2-(a-b)^2=4ab$ how would you solve for $a+b$?

[flat frigate]

Mosh

[fading cradle]

yeaaah

[pulsar condor]

Yeah isn't an applicable response to "how"

[fading cradle]

i mean how? i dont know

[quasi bison]

what's the original question?

[pulsar condor]

Pins

[fading cradle]

if a-b=5 and ab=-6 the what is a^4-b^4?

[quasi bison]

okay

so you know the values of two expressions

a-b and ab

$\underbrace{(a+b)^2}{\text{unknown}} - \underbrace{(a-b)^2}{\text{known}} = \underbrace{4ab}_{\text{known}}$

[flat frigate]

Ann

[quasi bison]

@fading cradle does that make it any clearer?

[fading cradle]

yes

[quasi bison]

are you able to answer mosh's leading question now?

[fading cradle]

ahhh. a+b=1

[pulsar condor]

(a+b)^2=1, yes

[fading cradle]

then how can i get a^4-b^4?

[pulsar condor]

wdym

you asked about a^2-b^2

[quasi bison]

the same strategy can be used to get the value of a^2 + b^2.

a^2 + b^2 - 2ab = (a-b)^2

[fading cradle]

how about a^4-b^4?

[quasi bison]

are you aware of the difference of squares identity?

once you know a^2 - b^2 and a^2 + b^2, getting a^4 - b^4 should be no issue.

[fading cradle]

sorry i have not fully understand factoring

[pulsar condor]

#❓how-to-get-help again

[quasi bison]

@trim imp what do you mean, no?

are we to understand this as a refusal to read the rules?

[pulsar condor]

They asked about Collatz

[torpid oxide]

Collatz is true

[safe radish]

@fading cradle Has your question been resolved?

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[solemn willow]

can i ask a question here?

[solemn willow]

.close

[safe radish]

Channel closed

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[kind roost]

is this channel occupied?

oh

nvm

[safe radish]

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[lean otter]

help please

[lean otter]

Digits 1, 2, ..., 9 are broken into three groups. Prove that the product of numbers in at least one of the groups is at least 72. 

anyone please

[steel stag]

any idea for what the product of each group would be if they had the same product?

[lean otter]

24

[steel stag]

is that 72/3?

like how what is 9! = 362880 evenly split between 3 products

x^3 = 9!

[lean otter]

yes

[steel stag]

oh here it's about multiplying, so 24+24+24=72 isn't really relevant

the total product is 1*2*3*..9

so if each group had the same product it would be the cube root of 9!

,w (9!)^(1/3)

[flat frigate]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

Requested by Plurmorant#1811

[steel stag]

does that make sense?

[lean otter]

why?

also, that number is not pretty

[steel stag]

yeah but it being near 72 is what's going to give us the answer 👀

so like you divide the numbers 1 to 9 into 3 groups and take their product

so like x*y*z = 9!

and if the groups have the same product it's just x^3=9!, and you cube root both sides

[lean otter]

does it mean that they can't be equal to 72??/

[steel stag]

not sure what you mean

1,8,9 has product 72

so here's the proof strategy here, what is the scenario where no product is at least 72, but the products multiply to as big a number as possible

just 71*71*71 right?

this is where no group reaches 72, but they're super close

[lean otter]

i don't understand really

i don't understand it well enough to write an answer

[steel stag]

do you know how contradiction proofs work?

like the opposite of at least one of the groups is at least 72 is that all groups have a product of 71 or less

[safe radish]

@lean otter Has your question been resolved?

[gloomy bone]

dynamic

[lean otter]

yes

[steel stag]

cool so we'll assume none of the groups have a product 72 or larger

what's their maximum product when you multiply all 3 together then?

71*71*71 right?

and the point is that if you multiply 1 through 9 you get a bigger number, so it's impossible

,w 71^3, 9!

[flat frigate]

Results provided by WolframAlpha

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

Requested by Plurmorant#1811

[lean otter]

yes

[lean otter]

oohhh

let me try and write this

@steel stag

it doesn't tell us what the product of one group has to be

[steel stag]

what do you mean?

[lean otter]

oh

here is my solution so far

Let’s assume that none of the groups can have a product greater than 72. Therefore, their maximum product has to be 71*71*71, which means that it must be equal to 9!.
    71^3 = 357911
    9! = 362880
    357911 is not equal to 362880; thus, the product of one group has to be 72 because ???

@steel stag

[steel stag]

I gtg, maybe someone else can help finish

[lean otter]

:(((

<@&286206848099549185>

Anyone please

Whyyyy no one help

well i have no idea how to solve this

[lean otter]

That doesn’t help lol

But thanks for trying to take a look

[bright crow]

You need to word it as a "contradiction".

[lean otter]

maximum possible value of abc + def + ghi = 620

1 x 2 x 3 + 4 x 5 x 6 + 7 x 8 x9

right?

[bright crow]

You showed that the product of all elements has to be less than 71³. But, the product of all elements is 9! which is greater than 71³ and hence contradiction.

[lean otter]

i think i got it

so assume it cant be 72

[bright crow]

Where did addition come into picture?

[lean otter]

three groups. just trying out differnet stuff

assume it cant be 72

it also cant be 71

since 71 is prime

so 70 is maximum product of abc

we have 3 such terms

abc + def + ghi

70 + 70 + 70 = 210

idk if it makes sense

[bright crow]

We just say product of elements of each group is ≤71 and that is ok

[bright crow]

Product of all the groups makes much more sense

[lean otter]

hmm yess

so 9! and 71^3

oh the previous guy did the same

yea that works

[safe radish]

@lean otter Has your question been resolved?

[lean otter]

I did

[lean otter]

But I did that

[bright crow]

So where are you stuck?

[lean otter]

How to prove that one group has to equal 72

[bright crow]

We don't have to prove one group has product of elements equal to 72. We just need that atleast 1 group has product in of its elements atleast 72.

[lean otter]

Yes

[lean otter]

Why

[bright crow]

Read your question again.

[lean otter]

Digits 1, 2, ..., 9 are broken into three groups. Prove that the product of numbers in at least one of the groups is at least 72. 

@bright crow

[bright crow]

Yes. We showed that every group cannot have product of their elements less than 72

[lean otter]

doesn't mean that at least one has to have a product of 72

[bright crow]

Atleast one group has to have product ≥ 72

[lean otter]

but that is not what the question is asking for

[bright crow]

What do you make of the last statement? "Prove that the product of numbers in atleast one of the groups is atleast 72"?

[lean otter]

but where am i proving that in my solution

[bright crow]

You can show that 72 is achievable

[lean otter]

how

[bright crow]

Choose 8,9 as the elements in one of the group

[lean otter]

you mean 8,9,1

[bright crow]

The question doesn't state that each group has three elements.

[lean otter]

oh, it doesn't

but they are saying in any formation of the groups

at least one has to have a product of at least 72

oh, i see

how is this solution

Let’s assume that none of the groups can have a product greater than 72. Therefore, their maximum product has to be 71*71*71, which means that it must be equal to 9!.
    71^3 = 357911
    9! = 362880
    357911 is not equal to 362880; thus, the product of one group has to be at least 72.

i understand now

[bright crow]

You showed that it is not possible for all three groups simultaneously to not have atleast 72 as product.

And you showed that group with 72 as product does exist

[bright crow]

If you are clear, remember to .close

[lean otter]

ok

is it good?

@bright crow

and then i will close

.close

[safe radish]

Channel closed

Closed by @fierce plaza

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[safe radish]

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[ashen badge]

how do i determine if f(x)=10 is bigO(x)

[safe radish]

@ashen badge Has your question been resolved?

[ashen badge]

<@&286206848099549185>

[steel stag]

uh depends on which bigO that is but 10 is less than f(x)=x when x>10

[ashen badge]

so f(x) = O(x)?

[steel stag]

f(x) is O(x) because it's smaller in the high x limit yeah

[safe radish]

@ashen badge Has your question been resolved?

[safe radish]

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[finite flint]

how does it turn to -sinx

[finite flint]

sin(pi-x) =sinx

how does it work when its sin(x-pi)

[pulsar condor]

sin(-(x-pi))=-sin(x-pi)

[safe radish]

@finite flint Has your question been resolved?

[rotund turtle]

your moving the graph right pi

which makes the graph be the same as -sinx

[finite flint]

okay thank you

.close

[safe radish]

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[safe radish]

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[short orchid]

Hello, so this is a physics question but it uses math and that is what I’m confused with.
So theta is a variable and we have been given these initial conditions. With this in mind, what would the max height be?

[short orchid]

I got an equation relating y and theta as $$y= 5 \tan(\theta)-\frac{5}{4} \sec^2(\theta)$$

[flat frigate]

floatoss

[short orchid]

Now I know how to proceed by using calculus, maximising the y. But the instructor proceeded with a way using discriminant

They converted the above equation into a quadratic in terms of tan(theta) and then found the discriminant in this way.

I don’t really understand what was done here

Ping me if someone replies, thanks

[safe radish]

@short orchid Has your question been resolved?

[short orchid]

<@&286206848099549185>

[safe radish]

@short orchid Has your question been resolved?

[outer estuary]

I've looked at this for like ten minutes and have no idea what was done here

Is this intermediate mechanics?

[short orchid]

idrk the stages, it’s not how it works in my country

it’s just mechanics for me

[outer estuary]

Gotcha, well I guess it doesn't matter. To solve for the highest point you'd take the derivative of the y function and then set your new function equal to 0 to find the turning point, but I can't tell from your diagram, is there a wall 5m away?

[short orchid]

yeah that’s a wall, and I don’t need the calculus way. I’ve already done it

there was this another way of doing it, via discriminant and that is what I’m confused with

[outer estuary]

Ah gotcha, well as far as what your teacher did I can't tell. As far as I know I'd just plug in the different possible values for time based on that distance with the horizontal component into your vertical component and optimize. glglgl :)

[short orchid]

Thanks. Using derivatives is better, but this algebraic way somehow is less calculative and that I am interested to know about it

[humble venture]

You can also complete the square (in terms of tan(theta)) to get the maximum

[safe radish]

@short orchid Has your question been resolved?

[safe radish]

Channel closed

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[short orchid]

.reopen

[safe radish]

✅

[short orchid]

well yeah, but that does not help with the way I'm asking about

[short orchid]

<@&286206848099549185>

[safe radish]

@short orchid Has your question been resolved?

[merry swift]

I can't say i know what your teacher did but i can suggest another algebraic way: completing the square on the highlighted quadratic in tan gives y=15/4-5/4(tan(theta)-2)^2. This is maximum when 5/4(tan(theta)-2)^2=0 giving y=15/4=3.75

oh sorry

i just read that someone else said that too lmfao

[short orchid]

bruh lol

[safe radish]

@short orchid Has your question been resolved?

[lean otter]

s=ut => 3 = 10costheta x t
at max height: v=0 => v^2 = u^2 + 2as => 0 😦 10sintheta)^2 + 2gH
s = ut + 1/2at^2 = 10sinthta (3 /(10 cos theta)) + 0.5 (g) (3 / (10cos theta)^2

and then relate perhpas to that eqn?
~~maybe im wrong 🥲 ~~

[short orchid]

idk how did you get 3 in there

but this is not how I need it, the thing is how did the instructor do the algebra

oh you mistook the 5 as 3, nvm. But yeah, it looks the same, but how do you bring up the discriminant thing now?

[short orchid]

.close

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[drowsy forge]

pretty much what i got but its wrong
ok change 300 to 380 but still got the wrong anwer
were my zeros worng
zeros i got were 2 and -9/8
the -9/8 wasnt reasonable so i did not use it so i used the interfal 2 and 0 instead

[safe radish]

@drowsy forge Has your question been resolved?

[mental minnow]

@lean otter Channel is occupied

[lean otter]

the disrespect

[mental minnow]

so you substituted p with 190?

[safe radish]

@drowsy forge Has your question been resolved?

[drowsy forge]

uhh wait

yea

i did

[mental minnow]

so what did you get as answer?

You should've actually gotten two answers and based on the two answers choose the one suitable that answered the initial question.

[drowsy forge]

isnt it just one answer

the answer i got was like 197

167*

167.04

[mental minnow]

when you have x^2 you always have 2 possible answers

of course though, one is not possible

[drowsy forge]

yea

i dont think surplus means you can have a negative number

[mental minnow]

i did the equation myself and unfortunately it's wrong. Try solving for x again mate

[drowsy forge]

ok

[mental minnow]

right. So you pick the positive of the two answers

[drowsy forge]

wait so my zeros are wrong?

i got (x-2)(9x+8)

[mental minnow]

I don't really understand the handwriting but it seems that you got (x-2) and (9x+8) as the two solutions right?

[drowsy forge]

yea

[mental minnow]

ok so you got your positive there

what is it?

[drowsy forge]

2

[mental minnow]

congrats

[drowsy forge]

yea i said that i have 2 as my zeros

[drowsy forge]

here

[mental minnow]

the two answers are 2 and -8/9

not -9/8

[drowsy forge]

oh yea my bad

well the 2 creates the intervals from 0 to 2

intergal

[mental minnow]

hope my answer allows you to finish solving the problem.

[drowsy forge]

eh not really

[lean otter]

The Mainboard or motherboard of a computer is a printed circuit board to which the components that make up the computer are connected, including the CPU (Processing Unit) and the Northbridge (connects the CPU with the other elements of the mainboard ).
The mainboard is made from fiberglass that is used to insulate the components.
A company that makes motherboards has designed a new type of card as shown:

After separating the CPU and Northbridge space on the mainboard, how much fiberglass, in u ^ 3, will be used to build a mainboard, if the board thickness is (⅓) ^ 3 ? (Express the result as a power)

[warped roost]

@lean otter stop posting in occupied channels.

[drowsy forge]

homie

lmao

[safe radish]

@drowsy forge Has your question been resolved?

[drowsy forge]

i just cant find the way to answer thie question everything always land back to 169

or just

something that doesnt wrok

it doesnt make snes

380-100int 0to 2(0.9x^2+0.3ln(1+x)

lol ive been stuck on this for like 4 hours now

[stiff inlet]

why are you using an ln?

[drowsy forge]

derivitive of a fraction

[drowsy forge]

oh yea i put this in wrong

it should look liket his

but it is still wrong so i have no idea hwa tim doing

[safe radish]

@drowsy forge Has your question been resolved?

[worthy hemlock]

Don't ask in an occupied channel

Read #❓how-to-get-help

[safe radish]

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[fallow nacelle]

How do I calculate the area of a triangle when given 3 sides?

[pastel hamlet]

a=1/2 b*h

oh sorry didn't see the 3 sides part lmao

https://www.cuemath.com/measurement/area-of-triangle-with-3-sides/

Does this help? @fallow nacelle

[worthy hemlock]

Heron's Formula

[fallow nacelle]

Will the triangle with sides 1.0, 1,5, 1.0 have an area of 0.49607837082461076?

[pastel hamlet]

That's what I got by plugging it in to a Heron's Formula calculator

[worthy hemlock]

I'm saying yes, because that looks like an online calculator value

[safe radish]

@fallow nacelle Has your question been resolved?

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[green skiff]

so for y(25)

would i replace all values of x with 25 then solve, or would i write y(x) as 25 then carry over the right side of the equation to the left?

[worthy hemlock]

Replace

[green skiff]

Thanks! So my answer would be 4.125

would this answer also be the same? since y(x) = height above the ground

wait, no

would the answer for y(25) be the answer to b.?

as answer A is asking for the horizontal distance

y(x) = vertical distance as stated from the word problem

is part b asking for a pair of coordinates?

or is it asking to explain the vertical intercept which is 2.5 in this case, and 2.5 is the height of the horshoe when the x value is 0

[tall saddle]

Do you remember what is a y-intercept?

[green skiff]

sorry

2.5 is the y intercept in this equation

so the explanation would be like:

"The numerical value of the vertical intercept is the height of the horseshoe when the x value is at zero"

or better:

"The value for the vertical intercept is 2.5, in context this means that the horseshoe has an initial height of 2.5 before throwing"

[safe radish]

@green skiff Has your question been resolved?

[green skiff]

sure

[safe radish]

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[knotty viper]

I don't understand what the domain means: 0 <= x < 1/4

1/4 of 2pi? so 1/2pi?

[safe radish]

@knotty viper Has your question been resolved?

[knotty viper]

.reopen

[quasi bison]

@knotty viper x is measured in radians and goes from 0 (inclusive) to 1/4 (exclusive) as written

the angle you are considering is 8πx, which goes from 0 to 2pi accordingly

does this answer your question?

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@knotty viper Has your question been resolved?

[safe radish]

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[gaunt fern]

how come my teacher multiplies both sides by ln

shouldnt he just chainrule sinx and e^x

[mental plover]

both ways work

[humble venture]

look up "logarithmic differentiation" if you want to know more about this method

[safe radish]

@gaunt fern Has your question been resolved?

[safe radish]

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[knotty viper]

Yes, thank you

[quasi bison]

...since you reopened this channel, do you have anything else to ask or is that it for now?

[knotty viper]

Yeah I have another question

This is the work I got so far, I don't know the next step

[quasi bison]

look closely at your last equation

sin(x) = ±4

what is the range of sin(x) for real inputs?

[knotty viper]

(0,360) ?

[quasi bison]

no

i'm talking about the range, not the domain.

if you graph y=sin(x), what values does it cover on the y-axis?

[knotty viper]

(-1,1)

[quasi bison]

[-1, 1].

-1 and 1 themselves ought to be included too.

[knotty viper]

ah okay

[quasi bison]

now, is 4 in this interval?

[knotty viper]

No

[quasi bison]

and is -4 in this interval?

[knotty viper]

No

[quasi bison]

so what can you conclude?

[knotty viper]

undefined?

[quasi bison]

no

you can conclude that the equation has no solutions.