help-10

[tawdry sail]

hello , is there a fast way to find the determinant of the equation

[tawdry sail]

i tried to create a matrix from this equation

but it was 4x3

so i thought that determinant would litteraly take a lot of steps

any advice

<@&286206848099549185> anyone

[tardy epoch]

you can just make an empty variable to make it 4x4

like (x,y,z,w)

[tawdry sail]

not sure howh would that make it easier

[tardy epoch]

it would make it easier to find the determinant

whatever "the determinant" means

if you have an actual problem statement, post the original question

[tawdry sail]

the equation is .
which one of the affirmations are true:

• if (m = 1, and a = 1) the equation have one unique solution
• if(m = a) the equation have infinite solutions
• if( m != a) the equation doesnt have any solution

the whole problem without the translation

[tawdry sail]

the determinant of a matrix

but not sure if its the way to go

[tardy epoch]

you don't even have a square matrix so why do you think you should use a determinant

[tawdry sail]

oh i forgot, for a moment i thought you can calculate the determinant of every matrix

@tardy epoch any idea how to solve this, been stuck for hour +

since the determinant isnt an option

[tardy epoch]

you are given options for a reason. use them

it's silly to show just the system of equations while ignoring the information in the question

and an absolute waste of helpers' time

[tawdry sail]

sorry for that,
how would i know which one has infinite solutions or not

[tardy epoch]

.

[tawdry sail]

my question was "after using the options , how can i conclure that a system has infinite solutions"
sorry if its a stupid question

anyways thx , i'll try to use the options

[obtuse pebble]

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[obtuse pebble]

Please don't occupy multiple help channels.

[quartz ore]

.close

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[vale echo]

k=(2^p)-1
N=(k^2)-1
prove that (2^p+1) is a factor of N

[obtuse pebble]

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[tawny tree]

why is it >= and not >

[worn yoke]

if a sequence is always greater than 1/4, but gets arbitratily close to 1/4, then its limit may be 1/4

[tawny tree]

I understand that however here there is some expression and not exactly a_n

[worn yoke]

we still have $\lim_{n\to\infty}a_n=\lim_{n\to\infty}a_{n+1}=g$

[warm shale]

cloud

[tawny tree]

so I can treat this expression as a limit on its own?

[obtuse pebble]

@tawny tree Has your question been resolved?

[tawny tree]

this probably just follows from the definition

[fossil crag]

g(1-g) is the limit of an(1-an+1)

since an goes to g

an+1 also goes to g

[tawny tree]

yes

[fossil crag]

and then operations on limits

(sum, difference, product, etc...)

[tawny tree]

ooh okay like that I get it now

[tawny tree]

yes that was in the lecture script

thank you

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[violet glacier]

hiiii everyone i'm having trouble with understanding gamma functions. how would you solve for (1/2)! any help is very much appreciated 🙏🙏

[zenith raft]

gamma(1/2), or (1/2)! ?

[violet glacier]

(1/2)!

[frigid burrow]

1/2 factorial?

[violet glacier]

Yes

[worn yoke]

so then we have (1/2)! = gamma(1/2 + 1)

[old lily]

did someone say gamma function

[violet glacier]

ohhhh i had a small error in my integration i just realized my mistake 😅

tysm guys ❤️

[violet glacier]

lollll yes

[obtuse pebble]

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[hybrid tartan]

How do I subtract in different bases such as 6345 in base 7 minues 1362 in base 7 and change numbers between different bases like 42 in base 10 when written in base 5 is

[hybrid tartan]

pls help

im kind of desperate

[opaque mesa]

the same way you do in base 10

you just have less numbers

1,2,3,4,5,6,10,11,12,13,14,15,16,20...

[obtuse pebble]

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[nocturne ore]

why the mid point between two numbers is (a+b)/2 ?

[latent walrus]

because thats the average of the numbers

[nocturne ore]

then question becomes why the average is (a+b)/2 ?

[timid silo]

if 0 < a < b, then u can define the midpoint as the point in the middle of segment ab between a and b. That is, midpoint = a + [(segment ab)]/2

m = a + (b-a)/2

Which simplifies to (a+b)/2

[nocturne ore]

why did we add half of b-a?

i mean, it makes sense intuitively but that doesn't mean i understand it.

[timid silo]

here's an example

basically midpoint = initial part of the segment + segment/2

[nocturne ore]

well

[timid silo]

Have u used vectors

[nocturne ore]

yes

[timid silo]

Ok this will be easier

that gives you an x coordinate between a and b

[nocturne ore]

let a,b,m element of R. a<b and m is the midpoint. thus segment am and segment bm must be equal.

[timid silo]

ok ur trolling 😭

[nocturne ore]

no im not

how am i trolling?

[timid silo]

am and mb

[nocturne ore]

why did you said that i'm trolling?

[timid silo]

too much stuff

[nocturne ore]

?

not sure i follow

[hidden compass]

By definition

Also, why would you want not to take the middle between the two numbers?

[nocturne ore]

i don't get why adding two numbers then dividing by 2 gives the half between them and i am trying to understand it

[hidden compass]

Let's say you have a < b

[viral blade]

the distance from a to m should be (b-a)/2, right

[hidden compass]

Yeah, (b-a)/2 is the distance between the midpoint and each of the endpoints

[nocturne ore]

nice

now i get it

a plus this

i was trying to understand linear interpolation but first i have to understand this

[nocturne ore]

okay this makes sense

thank you all. i'm done, how do i close this channel

[timid silo]

.close

[nocturne ore]

.close

[obtuse pebble]

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[nocturne ore]

.reopen

[obtuse pebble]

✅

[nocturne ore]

yes, thanks.

.close

[obtuse pebble]

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[nocturne ore]

.reopen

[obtuse pebble]

✅

[nocturne ore]

i will paint something

sorry wait

[crimson pike]

If this point is the mid point between am then yes

[nocturne ore]

yes it is

does it becomes a + (b-a)t if we want to generalize it?

when t is 1 it gives b, when t goes to infinity it gives a right?

[timid silo]

u can generalize it for any a,b no matter the order with vectors

the components are just A=<a,0>, B=<b,0>

the midpoint would be in M = A + AB/2

[timid silo]

.

[nocturne ore]

and if i want anywhere something in between it is A+AB*t where t is a percentage?

[timid silo]

yes

for example if t is 100

P = A + AB (100%) = A + AB(100/100)

P = A + AB

P = A + (B-A) = B

so the point is basically just B

[nocturne ore]

and this is called interpolation right?

[timid silo]

i have no idea

[nocturne ore]

okay thanks

[timid silo]

if u need it with numbers and not vectors you expand AB = B -A

P = A + ABt = a + (b-a)t

[nocturne ore]

i wasn't trolling by the way, sorry for the wrong impression. i didn't mean it

[timid silo]

dun worry

ok btw

other common form is P = a + (b-a)t = a + bt - at = a(1-t) + bt

[nocturne ore]

hold on

i have seen this formula 4 years ago

this one

a(1-t) + bt

it didn't make sense intuitively

[timid silo]

midpoint is this when t =50% and in one dimension

but yes it comes from P = A + ABt

it just means, start at A, then move along the segment AB from 0% to 100% (0 to 1)

[nocturne ore]

what does a(1-t) + bt mean

i know that they are same but

[timid silo]

a bunch of annoying terms that are hard to read

[nocturne ore]

construction of desired point is different

[timid silo]

if you rewrite it as a + (b-a)t is easier to notice that its just P = A + ABt

in vectors AB = B-A

so AB = <b>-<a>

[nocturne ore]

im checking this right now

what is v?

[timid silo]

A vector that points from r0 to r1

[nocturne ore]

r1-r0

?

[timid silo]

(that is r0r1 = r1-r0)

Yes

[nocturne ore]

https://cdn.discordapp.com/attachments/903430332324405288/1296957405967548476/hq720.png?ex=67142d35&is=6712dbb5&hm=ee673c2c9e964ffea8891496aa46febf27b3f6204bcf0c5e0174b3c6d903a704&

what does (1-t)r0 mean?

[clear cosmos]

Hope this helps

Take this

[nocturne ore]

okay what do we get by multiplying 1-t with r0

.close

[obtuse pebble]

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[timid silo]

(1-t)r0 is the equivalent of (1-t)a

just in the ugly way

P = (1-t)a + tb = A + ABt = a + (b-a)t

[nocturne ore]

.reopen

[obtuse pebble]

✅

[nocturne ore]

(1-t)a what would this mean?

can you explain it in line numbers?

[timid silo]

nothing

just 'simplification' that is not even a simplification. it just makes it harder to read

the nice formula is P = A + ABt

[timid silo]

or P = r0 + r0r1 t (the same, whe just changed the name of the vectors)

no, it doesnt mean anything

[nocturne ore]

i know it is nice

but surely it must mean something

[timid silo]

if u want to interpret it t would be the % of how much we take of a and what we dont take of b)

(1-t)a + (t)b

if t=0=0%

100% of a + 0% of b

if t = 1 = 100%

0% of a + 100% of b

if t = 0.5 = 50%

50% of a + 50% of b (the formula would simplify to 0.5a + 0.5b)

[nocturne ore]

isn't it (1-t)a + bt

[timid silo]

no

reagroup terms, u will notice, a + (b-a)t

oh yes

bruh

[nocturne ore]

but it is here

[timid silo]

ye forget it

[nocturne ore]

okay now im reading it again

[timid silo]

t means the % you move in the segment

[nocturne ore]

damn

[timid silo]

and how close you are to the endpoints

[nocturne ore]

this is neat.

how do you know so much

[timid silo]

jokes on you i dont even know what is interpolation

[nocturne ore]

i mean how did you come up with this

[timid silo]

i didnt even type what i mean

but still similar idea

(1-t)a + tb read in percentage way

(100%-30%)a + (30%)b as an example (because 1 = 100/100 = 100 (1/100) = 100%

70%a + 30%b

if you think about this, 50%a + 50%b being average would intuitively lead that 70%a + 30%b means that a is weighted more than b

so the number should be to the left side of the midpoint

30% of the entire segment starting from a to b to be more precise

[nocturne ore]

linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points i.e. what we did right now

t is f(t)=t but doesn't have to be

[nocturne ore]

not gonna lie, i understand it vaguely, not precisely.

anyways, i gotta go. thanks for everything.

.close

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[south barn]

given: AB=AC=AD
∠BCD=140

∠BAD=???

[obtuse pebble]

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[violet ingot]

why isnt this correct? both of my m and n values are perfect sqaures here and 36 is not a perfect sqaure

[obtuse pebble]

@violet ingot Has your question been resolved?

[shut lagoon]

6*6 = 36

Moreover, even if that was not the case, it would not be a counter example to the statement above.

A counter-example has to be m,n such that mn is a perfect square but m or n are not.

[violet ingot]

so to prove it false, all i have to do is use 1 non-primes that make a prime like 1 and 4 and pick the 3rd option?

[shut lagoon]

I'm not sure what you mean by 2 non-primes that make a prime.

1*4 is 4, which is a perfect square, but both 1 and 4 are perfect squares, so it's not a counter example.

[violet ingot]

1,3 my bad got confused with a seperate problem

[shut lagoon]

But then 1*3 is not a perfect square

[violet ingot]

ah

[shut lagoon]

The exercise is to factor some perfect square into non perfect square factors

[violet ingot]

oh

[violet ingot]

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[red cipher]

Hi just curious is it possible to simplify that further with the factorials

[timid silo]

where

[red cipher]

Bottom right

[teal turret]

Yea

[timid silo]

u got 1=1

[teal turret]

Can u expand r!

@red cipher

[red cipher]

Right OK I see

Thx

.Close

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[steel lagoon]

how did the radical turn negative here?

[steel lagoon]

i know it has something to do with x approaching negative infinity but how does that work?

[unreal musk]

As $\sqrt{x^2} = \abs{x}$, if $x$ is negative, you have $\sqrt{x^2} = -x$

[warm shale]

@unreal musk

[hot dawn]

Erm are u sure

Isn’t this just a+b = -(a-b)

In the denominator

[unreal musk]

(that isn't true either, but it isn't)

(also hiii )

[hot dawn]

Hi

[hot dawn]

I’m pre sure all they did was factor out the negative sign in the denominator but ok

[steel lagoon]

so i would multiply each root by -1 whenever x approaches negative infinity

[light flume]

intuitively

your expression in the first line will be negative

when u multiply by 1/x / 1/x

the x goes into the square root but u lose the information for the fraction being negative

[steel lagoon]

thats not it

[unreal musk]

There isn't initially a negative sign to begin with, if you divide numerator and denominator by x, you get 6 as the numerator, but to be able to "absorb" the x into the root, you'd write it as -sqrt{x^2}

[hot dawn]

Oh I’m stupid whoops

[unreal musk]

Which is where the negative signs come from

[steel lagoon]

when i multiply by 1/x my expression is negative?

[light flume]

multipy by 1/x / 1/x

as in

(1/x)/(1/x)

ok just before u even evaluate anything and stare at the first line

you know the denominator is going to be positive at least

so your overall expression after limit has to be negative

when you divide top and bottom both by x

and absorb the x into the square root

u are losing 'information'

[steel lagoon]

i see

thanks

just tell me this please, mechanically, if i just multiply by -1 next time x approaches negative infinity will it work out

[light flume]

case by case scenario

...

[restive acorn]

Basically the argument you're using here is this:

You know that 1/x is negative

Therefore, when you multiply (1/x) *rad(expression)

To account for the fact that this number should be negative,

We write it as -rad(expression/x^2)

Because otherwise rad(expression) is positive

[steel lagoon]

that makes sense

thanks

is that the case only for radical expressions though

[restive acorn]

Well even roots specifically

[steel lagoon]

because on the top when i multiplied 6x by 1/x it wasn't negative

[restive acorn]

Probably not the ONLY time but this is specifically an issue for even roots

Because even roots are always positive for real numbers

So when you're expecting a negative number

If you follow the "normal" algebra

You'll lose that information as the phrase goes

[steel lagoon]

thank you

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[gentle topaz]

How to solve this problem?

[gilded needle]

look at rows 3 and 5

[gentle topaz]

I guess the answer is A but i am not sure

[restive acorn]

Nvm

The last column is the solution column

Since they called it an augmented one

[gilded needle]

your 3rd row is saying 3y = 4 and your 5th row is saying -3y = 2

assuming we call the variable associated with the second column y

[restive acorn]

Do you know how to read the row and write the equation for it?

[gentle topaz]

yea

[restive acorn]

Try it for row 3 and 5 as Bungo said

What would those rows tell us as equations

[gentle topaz]

R3 and R5 would be all 0

[restive acorn]

The first column tells me about the first variable

For example

The first row says

5x + 5y + 5z = 4

Or

5x1 + 5x2 + 5x3 = 4

Two common ways to name 3 variables

The second equation says

4x + 4y + 3z = 4

From the second row

What equation does the third row represent?

[gentle topaz]

3y=4

[restive acorn]

Solve for y

[gentle topaz]

4/3

[restive acorn]

Then check what the 5th equation says

[gentle topaz]

-3*4/3=2

[restive acorn]

No no

It doesnt tell you solutions

It tells you coefficients on the variable

Write it as y

Well

[restive acorn]

This is fine too

You see that this is a contradiction

That's because the 5th equation implies

y = -2/3

Those 2 equations are giving two different values for the same y

[gentle topaz]

so it's no solu

[restive acorn]

Yes

Exactly

[gentle topaz]

thank you

[restive acorn]

Yw

[gentle topaz]

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[obtuse pebble]

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[last shell]

given the diameter of a circle and the length of a chord, how would I find the distance from the center of the circle to the chord?

[light flume]

have u tried drawing a diagram to start?

if so, try constructing triangles with the chord and the centre of the circle

[last shell]

Will do ty

So radius²=(1/2 chord length)²+ distance²

[light flume]

yeah

u can rearrange and solve pretty easily then

[last shell]

Yeah, ty

[light flume]

nw

[last shell]

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[gloomy vector]

find the value of $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\dots+\frac{22}{20!+21!+22!}$

[warm shale]

Skill_Issue

[gloomy vector]

im fucked 😭

[versed stratus]

Where is this from

[gloomy vector]

ans choices are
1/2!-1/22!
1/3!-1/22!
1/2!-1/21!
1/3!-1/21!
1/4!-1/22!

[gloomy vector]

seamo 2024 (it just finished like half an hour ago i just wanna cross check)

[river falcon]

its actually very easy

write the general term

uh

[gloomy vector]

or like n/((n-2)!+(n-1)!+n!)

actually this is wrong nvm

[river falcon]

$T_n = \frac{n+2}{n! + (n+1)! + (n+2)!}$

[warm shale]

Astar777

[river falcon]

now take n! common

[gloomy vector]

$T_n = \frac{n+2}{n! (1+(n+1)+ (n+1)(n+2))}$

[warm shale]

Skill_Issue

[river falcon]

yes

simplify

[gloomy vector]

$$T_n = \frac{n+2}{n! (n^2+4n+4)}$$
$$T_n = \frac{1}{n! (n+2)}$$

[warm shale]

Skill_Issue

[river falcon]

yep

now multiply and divide by (n+1)

[gloomy vector]

$$T_n = \frac{(n+2)-1}{(n+2)!}$$

[river falcon]

(n+2)-1

(numerator)

[warm shale]

Skill_Issue

[gloomy vector]

oh

1/(n+1)!-1/(n+2)!

[gloomy vector]

telescope its 1/2!-1/22! right

[river falcon]

yes

[river falcon]

yes

[gloomy vector]

woohoo

guessed randomly and it was right

.close tyy <3

[obtuse pebble]

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[rose scroll]

There is a scammy way to do this, if you choose n=1

Clearly, this problem has an induction version

and option 1 is the only pair that gives the answer

[obtuse pebble]

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[grand yoke]

ax^2+bx-45 = c(x-5)(x+3)

compare coefficients after expanding

@sinful sonnet

[sinful sonnet]

What is the formula for this

[grand yoke]

theres no formula

[sinful sonnet]

Ohh

[river falcon]

this? this is for finding roots

but you already have roots

so you can do wht death.cv said

[obtuse pebble]

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[obtuse pebble]

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[signal glen]

hello

how do i solve this?

i know i can use the two points in a lagrange formula perhaps?

does P(n) = n+3 having 2 solutions suggest its a quadratic?

[obtuse pebble]

@signal glen Has your question been resolved?

[signal glen]

<@&286206848099549185>

[obtuse pebble]

@signal glen Has your question been resolved?

[rigid skiff]

what does p(n) = n+3 mean here

[slate hamlet]

First find the polynomial using the given solutions

Then you get another polynomial P(n) = n + 3

Well

It might not be a quadratic

They specified two distinct integer solutions

[rigid skiff]

what hurts in asuuming

so let the polynomial be

ax^2+bx+c = 0

from the question

[slate hamlet]

Then it's solvable fairly easily

[rigid skiff]

289a+17b +c = 10

sorry

and 576b+24b+c = 17

[gritty vessel]

how do you know that the function is quadratic though

[slate hamlet]

576a

[rigid skiff]

asuume for now

if it wont work

try smthn else

subtract 1 from 2 to get

[gritty vessel]

just because the function has two integer solutions doesn't mean it can't have other non-integer solutions

[slate hamlet]

If you assume it's quadratic you'll get a answer

[rigid skiff]

287a +7b = 7

or 41a+b=1

[gritty vessel]

yes but that's terrible mathematics because it presupposes something that you don't necessarily know is true

[rigid skiff]

yes but rn there is no other option

u wanna be stuck up here thinking about it

like it has a billion other roots we dont know

[gritty vessel]

the phrasing of the question implies that there is only one possible value of the product

[rigid skiff]

mathematics is about assumptions

and proving wether they are correct or not

[gritty vessel]

no math is about proof

[rigid skiff]

we shall first try this

[slate hamlet]

Wait

[rigid skiff]

if we reach a dead end

we will try something else

[rigid skiff]

what if this was time bound

[slate hamlet]

The condition for integer roots is that all the coefficients must be a multiple of each other

[rigid skiff]

u wanna prove every other root exists

wise ahh goat

[gritty vessel]

don't see why that changes anything in principle

[rigid skiff]

ok , we can try something else

i see asuming has faulterd

[obtuse pebble]

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[dusk widget]

#1260424130822541403 message

[dusk widget]

hi Layla, before you come in and say "hi Kepe"

[dusk widget]

but yes, I need this proof checked over, if anybody is willing to 🙏

[obtuse pebble]

@dusk widget Has your question been resolved?

[dusk widget]

ig I can wait

.close

[obtuse pebble]

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[obtuse pebble]

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[low prairie]

Hello, I have a reflection regarding Dirac's distribution. Given that a distribution is related to a scalar product and that integration is a linear operation, can we deduce that Dirac's distribution is an element of the dual space of the test function space? Additionally, based on the definition commonly used by physicists and engineers, we can observe that the distribution yields a scalar.

[low prairie]

,tex $\int_{-\infty}^{\infty} \delta(x) , f(x) , dx = f(0)$

[warm shale]

hisitis__

[obtuse pebble]

@low prairie Has your question been resolved?

[obtuse pebble]

@low prairie Has your question been resolved?

[low prairie]

<@&286206848099549185>

[obtuse pebble]

@low prairie Has your question been resolved?

[low prairie]

.close

[obtuse pebble]

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[obtuse pebble]

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[mellow jewel]

for b

how do we know which region they want?

aren't both possible regions?

[timid silo]

prob rightside

[red dock]

This same argument can be made about the right side

[timid silo]

dam its right (i mean ur message its true)

okay maybe its the rightside because each curve is intersected with each other JUST once (yes the way it was given its kinda ambiguous somehow)

[red dock]

Yea it's a very ambiguous loosy goosey 🦆 question

They should have mentioned like y>= 0 instead of "bounded by y=0"

"y>= 0, y<= lnx, y<=4-x^2 " something like that

[obtuse pebble]

@mellow jewel Has your question been resolved?

[obtuse pebble]

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[austere bane]

Is this correct?

[glass dagger]

What's wrong with the first graph?

[obtuse pebble]

@austere bane Has your question been resolved?

[austere bane]

Is this not F(X) = - root(X)

I was going by the logic if I replace X by -ve X then it would be incorrect

[glass dagger]

-root(X) takes on negative values but the first graph stays above the x axis

[obtuse pebble]

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[dapper robin]

Help

[dapper robin]

Can this be solved thinking them as addition of angles

Without using the addition of inverse cotangent addition property

Maybe we can establish a pattern for 4k^2 + 3

Then analyse it

<@&286206848099549185>

.close

[obtuse pebble]

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[fallow cloud]

Help

[fallow cloud]

How do I mathematically calculate a number to make it tenths, so like 24,385638 -> 24,3

Help pls

.close

[obtuse pebble]

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[sacred sedge]

how do u do thois

[timid silo]

Just try values for x

[worn yoke]

$h^{-1}(2)$ is whichever value of $x$ satisfying $h(x) = 2$

[warm shale]

cloud

[obtuse pebble]

@sacred sedge Has your question been resolved?

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[foggy merlin]

im super lost on this question

[foggy merlin]

is my work right so far and if yea how do i continue

[obtuse pebble]

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[open spoke]

4a tips?

[tardy epoch]

Do you know the formula for A * a?

Say for N=3

[obtuse pebble]

@open spoke Has your question been resolved?

[open spoke]

3 x 2 * 2 x 1?

Not sure what a's dimensions are exactly

My bad for afk

[tardy epoch]

a is (a1, a2)

[open spoke]

(a1, a2)^T right?

as a column vector?

Otherwise it wouldn't make sense

So what does that mean for us?

[tardy epoch]

Sure

[tardy epoch]

.

[open spoke]

It's of this form

And for each b_i we have

I just tested it with N=3

[tardy epoch]

Well b is an N x 1 vector yes

That doesn't really answer my question

[open spoke]

That is the formula for A * a

[tardy epoch]

It's not

[open spoke]

oh i think

[tardy epoch]

I recommend reviewing matrix multiplication

[open spoke]

a11 a12
a21 a22
a31 a32

is A right

a1
a2

is a

b1
b2

is b

[open spoke]

Then it becomes

a11a1 + a12a2
a21a1 + a22a2
a31a1 + a32a2

[tardy epoch]

Yea

[open spoke]

Ohh i think i see what we must do, do we basically try to get all of that to be the same as what f(a) is defined as?

[tardy epoch]

N = 3 so your b needs how many variables?

Yes

[open spoke]

3

Meaning of life

Yes, are you familiar with the problem ?

[tardy epoch]

Guess Aspect got this

GL

[open spoke]

Nevermind, I think I know how to do it. Thank you though

[obtuse pebble]

@open spoke Has your question been resolved?

[obtuse pebble]

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[timid silo]

A water tank develops a leak. Each week, the tank loses 7 gallons of water due to the
leak. Initially, the tank is full and contains 4500. Using GRESA method.
a. How many gallons left 4 weeks later?
b. How many weeks until the tank is half-full?
c. How many weeks until the tank is empty?

[dusk widget]

GRESA - Given, Required, Equation, Solution, Answer?

anyways, you should approach this problem by setting up an equation that describes how much water the tank has after t weeks

[timid silo]

They wanted the given like this type.

example: 𝑎4 = 20, 𝑎9 = −15, 𝑎2024 =?

[dusk widget]

hmm

[dusk widget]

well, we should start with this anyhow

[timid silo]

ok

[dusk widget]

at 0 weeks, we have 4500 gallons. after 1 week, we’re at 4493 gallons. after 2 weeks, we’re down to 4486 gallons, and so on

[timid silo]

a) Given: - 7 gallons of water, + tank is full and contains 4500, = 4 weeks later

[dusk widget]

can you write down an equation that describes how much water we’re at after t weeks?

[timid silo]

uhh...

[dusk widget]

let’s call the amount of water the tank has at a given time W

and this number is going to depend on the amount of time, t, that’s passed in weeks

[dusk widget]

this is your hint

I’ll start you off

[gloomy vector]

2ml of leakage per minute lmao

[dusk widget]

when W = 4500, t = 0

when W = 4493, t = 1

when W = 4486, t = 2

and so on

the water is being lost at a rate of 7 gallons a week, so as t increases, W should decrease

what’s the equation we need to write to describe this?

[timid silo]

How did you do that-

[dusk widget]

do what?

[timid silo]

this

[dusk widget]

well, we know that after each week, the tank has lost 7 gallons of water

[timid silo]

yeah?

[dusk widget]

so when 2 weeks have passed, it’s lost 14 gallons

4500 - 14 = 4486

[timid silo]

OHHHH

[dusk widget]

in general, the equation is going to look like

(Water after t weeks) = 4500 - ??

what goes in the ??

[timid silo]

So I have to do this until I reached 4 weeks?

[dusk widget]

yeah, but you should use an equation rather than subtracting 7 each time

imagine subtracting 7 until the tank is half full

that would take us like

an hour!

[timid silo]

oh

How would do that

[dusk widget]

see here

we need to write down an equation to describe the amount of water in the tank after t weeks, to make our lives easier

[timid silo]

4 = 4500 - 7?

[dusk widget]

oh no

it can't be 4

and it's also not 7

[timid silo]

oh

[dusk widget]

on the left hand side of the equation, we have the amount of water in the tank at time t

W(t), if you've seen this notation before

W(1) = water in the tank after 1 week
W(2) = water in the tank after 2 weeks
...
W(t) = water in the tank after t weeks

as a result, the right hand side of the equation should depend on t

because the water in the tank is intrinsically tied to the amount of time elapsed

[timid silo]

So I have to use this method?

[dusk widget]

well, that was to give you ideas

subtracting 7 all the way down is really slow and annoying

[timid silo]

oh

[dusk widget]

we'd like to do better

W(t) = 4500 - ???

the ??? should depend on t, because the left hand side depends on t

[dusk widget]

I promise, once we have this, our lives will be so much easier

[timid silo]

time... and week(time)

..

[dusk widget]

maybe I'll write this out again, more suggestively

[timid silo]

So... time is missing!

[dusk widget]

yes!

somehow, the ??? needs to contain time

[timid silo]

what's W?

Week?

[dusk widget]

the amount of water in the tank at time t

W for water

[timid silo]

Ohhh

[dusk widget]

so we know that W(0) = 4500, right?

[timid silo]

then 7 is the water?

[dusk widget]

7 gallons per week

[timid silo]

Yeah

[dusk widget]

okay

so now W(0) = 4500 - what?

I promise this is relevant

[timid silo]

7(1) = 4500

[dusk widget]

that seems a little off

W(0) = 4500

after 0 weeks, we have 4500 gallons in the tank, yeah?

nothing has leaked out yet

[timid silo]

Yes?

[dusk widget]

awesome

now let's plug 4500 into our equation, even though we don't know what it is yet

4500 = 4500 - ???

what should ??? be, in this case?

[timid silo]

??? is time and we don't have time

therefore... 0?

[dusk widget]

yeah

it's 0

good!

[timid silo]

4500 = 4500 - 0

[dusk widget]

now, after one week, we know that the tank loses 7 gallons, right?

so it's at 4493 gallons remaining

[timid silo]

Yes

[dusk widget]

down from 4500 a week ago

nice

now let's plug it in

4493 - 4500 - ???

what's ??? now?

[timid silo]

How did you get that answer?

[dusk widget]

4500 gallons - 7 gallons = 4493 gallons

[timid silo]

Oh okay

[dusk widget]

right, so this

[timid silo]

We still don't have time

Therefore, 0

[dusk widget]

nono, there's been a misunderstanding

1 week has passed

1 week, 7 gallons

[timid silo]

OHHH CONVERT 1 WEEK TO TIME

[dusk widget]

we don't even need to think about weeks or gallons, technically

[dusk widget]

we just need to solve this equation for ???

and keep doing this until the pattern is obvious

that's how we'll find what ??? is in general

[timid silo]

Okay so we have 1 week now. therefor, Convert 1 week to hours to get the time since it's just dates

[dusk widget]

agh

I don't think that's what you want to do

weeks are a unit of time here

nothing to convert

[timid silo]

oh

[dusk widget]

t = 1 means 1 week has passed

[timid silo]

4493 - 4500 - 7?

[dusk widget]

yes!

so after 1 week, ??? is equal to 7

what about 2 weeks?

[timid silo]

Shouldn't this be 4493 = 4500 - 7?

[dusk widget]

yes

you're correct

??? = 7, when t = 1

[timid silo]

14?

[dusk widget]

and we saw earlier that ??? = 0, when t = 0

[dusk widget]

yes!!

so ??? = 14, when t = 2

now, what is ???

[timid silo]

time

[dusk widget]

it clearly depends on what t is

[dusk widget]

it depends on time

it isn't time itself

[timid silo]

the week?

[dusk widget]

I want you to write something like ??? = at, where t is the number of weeks elapsed, and a is a number you should find

that is, W(t) = 4500 - at

what is a?

[timid silo]

....

[dusk widget]

a is a number, but what number should it be?

[timid silo]

atmosphere...?

oh

okay

[timid silo]

7?

[dusk widget]

‼️

well done!

now we have an expression that tells us the amount of water in the tank after t weeks

W(t) = 4500 - 7t

[timid silo]

okay

[dusk widget]

so after 4 weeks, what do we have?

4 weeks means we want to find W(4)

and we can do that by plugging in t = 4

[timid silo]

W(4) = 4500 - 7(4)

[dusk widget]

very well done!

now, what is that number?

just use a calculator or something

[timid silo]

W(4) = 4500 - 7(4)
W(4) = 4500 - 28
W(4) = 4472

[dusk widget]

awesome

[timid silo]

That's the answer?

[dusk widget]

indeed!

the tank has 4472 gallons of water left, after 4 weeks

congratulations!

you did part a)!

[timid silo]

A water tank develops a leak. Each week, the tank loses 7 gallons of water due to the
leak. Initially, the tank is full and contains 4500. Using GRESA method.
a. How many gallons left 4 weeks later?
Given: - 7 gallons of water, + tank is full and contains 4500, = 4 weeks later
Requirement: W(4) is the number of gallons left 4 weeks later
Equation: W(t) = 4500 - 7t
Solution: W(4) = 4500 - 7(4)
W(4) = 4500 - 28
W(4) = 4472
Answer: There are 4472 gallons left 4 weeks later.
b. How many weeks until the tank is half-full?
c. How many weeks until the tank is empty?

[dusk widget]

yay!

part b, now

[timid silo]

wait

What do I write in the requirement and equation?

[dusk widget]

W(4) is required

the equation is probably, if I had to guess, W(4) = 4500 - 7t?

I'm not too familiar with this method of writing an answer, my apologies

[timid silo]

correct?

[timid silo]

It's fine

[dusk widget]

sure thing, but I would probably mentioned under "Required" that W(4) is the number of gallons left, 4 weeks later

so

[timid silo]

Okay

[dusk widget]

ah, hold on

[timid silo]

Hm?

[dusk widget]

Equation: W(4) = 4500 - 7t

it should be W(t) = 4500 - 7t

not W(4)

W(4) is 4500 - 7(4)

[timid silo]

Ah okay

[dusk widget]

[timid silo]

correct?

[dusk widget]

looks good to me

ready to do part b?

[timid silo]

Okie lemme write it and we can do part b

[dusk widget]

no problem

[timid silo]

A water tank develops a leak. Each week, the tank loses 7 gallons of water due to the
leak. Initially, the tank is full and contains 4500. Using GRESA method.
a. How many gallons left 4 weeks later?
Given: - 7 gallons of water, + tank is full and contains 4500, = 4 weeks later
Requirement: W(4) is the number of gallons left 4 weeks later
Equation: W(t) = 4500 - 7t
Solution: W(4) = 4500 - 7(4)
W(4) = 4500 - 28
W(4) = 4472
Answer: There are 4472 gallons left 4 weeks later.
b. How many weeks until the tank is half-full?
c. How many weeks until the tank is empty?

We can procceed

[dusk widget]

part b wants you to find the amount of time it takes until the tank is half full

that is, we wanna know when W(t) = 0.5 * W(0)

0.5 * W(0) because the tank is full at 0 weeks, and 0.5 cuts that number in half

[timid silo]

Ah okay

[dusk widget]

that is, W(t) = 0.5 * 4500 = 2250

so "when W(t) = 2250, what is t equal to?" is the question we'd like to answer for part b

does that make sense?

[timid silo]

Yes

[dusk widget]

fantastic

remember how W(t) = 4500 - 7t?

[timid silo]

Yes

[dusk widget]

we now also have W(t) = 2250

since 2250 and 4500 - 7t are both equal to W(t), we can equate them

like this -> 2250 = 4500 - 7t

[timid silo]

Where did you get 2250

[dusk widget]

tank is half full means W(t), the water in the tank at time t, is equal to 0.5 * W(0), since W(0) is the tank at full capacity

but we know that W(0) = 4500, so W(t) = 0.5 * 4500 = 2250

[timid silo]

I see

[dusk widget]

now it's just a matter of solving this equation for t

do you see how our work earlier is paying off?

[timid silo]

W(t) = 0.5 * 4500 = 2250
2250 = 4500 - 7t

[dusk widget]

we don't need to subtract 7 a bunch of times

[timid silo]

BWAHAHAHA Yeah

[dusk widget]

we just solve for t and get our answer

so, 2250 = 4500 - 7t

can you solve this for t?

[timid silo]

W(t) = 0.5 * 4500 = 2250
2250 = 4500 - 7t
2250 - 4500 = - 7t
-2250 = -7t
divide both sides
321.428571429 = t

[dusk widget]

looks good!

but, I suspect we don't really want a decimal week

does the problem only want whole numbers?

[timid silo]

W(t) = 0.5 * 4500 = 2250
2250 = 4500 - 7t
2250 - 4500 = - 7t
-2250 = -7t
divide both sides
321 = t

[dusk widget]

ah, you actually want 322, I think

because after 321 weeks, the tank still has a little more than half its water left

whereas after 322, it's less than half

you could also just leave it as t = 321.428571429, but idk how well your teacher will take this answer

maybe they take it very well, maybe not

idk

[timid silo]

If we tried to round up till will look like this
321.43143

[dusk widget]

mhm

[timid silo]

So it's still 321

[dusk widget]

well yeah, about 321 or 322 is the correct answer

it's just a little ambiguous rn, cause idk if your class accepts decimals answers

if they do, we can just move on

if not, we'll need to change the answer to either 321 or 322

[timid silo]

So that's the answer?

[dusk widget]

yes

up to decimals, it is correct

[timid silo]

A water tank develops a leak. Each week, the tank loses 7 gallons of water due to the
leak. Initially, the tank is full and contains 4500. Using GRESA method.
a. How many gallons left 4 weeks later?
Given: - 7 gallons of water, + tank is full and contains 4500, = 4 weeks later
Requirement: W(4) is the number of gallons left 4 weeks later
Equation: W(t) = 4500 - 7t
Solution: W(4) = 4500 - 7(4)
W(4) = 4500 - 28
W(4) = 4472
Answer: There are 4472 gallons left 4 weeks later.
b. How many weeks until the tank is half-full?
Given: - 7 gallons of water, + tank is full and contains 4500, the tank is half-full = 0.5
Requirement: t when W(t) = 2250
Equation: W(t) = 0.5 * 4500 = 2250
Solution: W(t) = 0.5 * 4500 = 2250
2250 = 4500 - 7t
2250 - 4500 = - 7t
-2250 = -7t
divide both sides
321.43143 = t
Answer: There will be 321.43143 weeks until the tank is half-full.
c. How many weeks until the tank is empty?

[timid silo]

Correct?

[dusk widget]

honestly, it might be good enough, but I'd use 322 instead of 321 as the answer, if I had to round

also, the requirement is probably smth like "t when W(t) = 2250"

[timid silo]

Is everything correct?

[dusk widget]

W(t) = 2250, not 225

and I still feel like 322 is the better answer, but if you insist, we can accept 321

[dusk widget]

referring to this, btw

Requirement: t when W(t) = 225

[timid silo]

But the logical fact is the decimals doesn't round up to 322

[dusk widget]

you're correct about rounding, but that's technically not what we care about here

we care about when the tank is actually at half capacity

and when t = 321, the tank is still a little bit over half capacity

at t = 322, it's a little bit under

so t = 322 is the first whole number for which the tank is actually less than half full

[timid silo]

They are both the same when it comes to decimals

[dusk widget]

I'm pretty sure we do not really care about decimals in this problem

we care about when the tank is at less than half capacity

[timid silo]

at the same time both have problems

1 is under

1 is over

They both have problems towards the amount of the tank if we wanted whole numbrs

[dusk widget]

of course, but 322 satisfies the contraint of being the first number for which the tank is actually at half capacity

321 does not

of course, none of this matters if we could just use t = 321.428571429

[timid silo]

Bruh ykw I'm rounding up and still keeping the decimal

[dusk widget]

I mean, it is your choice in the end haha

I don't really know what the teacher wants

I would just use t = 321.428571429, honestly

[timid silo]

Were literally debating satisfaction and logical analysis of the half capacity

[dusk widget]

yeah, maybe we ought to just move on

[timid silo]

A water tank develops a leak. Each week, the tank loses 7 gallons of water due to the
leak. Initially, the tank is full and contains 4500. Using GRESA method.
a. How many gallons left 4 weeks later?
Given: - 7 gallons of water, + tank is full and contains 4500, = 4 weeks later
Requirement: W(4) is the number of gallons left 4 weeks later
Equation: W(t) = 4500 - 7t
Solution: W(4) = 4500 - 7(4)
W(4) = 4500 - 28
W(4) = 4472
Answer: There are 4472 gallons left 4 weeks later.
b. How many weeks until the tank is half-full?
Given: - 7 gallons of water, + tank is full and contains 4500, the tank is half-full = 0.5
Requirement: t when W(t) = 2250
Equation: W(t) = 0.5 * 4500 = 2250
Solution: W(t) = 0.5 * 4500 = 2250
2250 = 4500 - 7t
2250 - 4500 = - 7t
-2250 = -7t
divide both sides
321.43143 = t
Answer: There will be 321.43143 weeks until the tank is half-full.
c. How many weeks until the tank is empty?

[dusk widget]

empty...

[timid silo]

So is everything correct?

[dusk widget]

same thing as in part b!

[timid silo]

WAITTT

[dusk widget]

waiting!

[timid silo]

Is everything correct?

[dusk widget]

seems good to me

[timid silo]

Okie lemme write it down rq

[dusk widget]

[timid silo]

A water tank develops a leak. Each week, the tank loses 7 gallons of water due to the
leak. Initially, the tank is full and contains 4500. Using GRESA method.
a. How many gallons left 4 weeks later?
Given: - 7 gallons of water, + tank is full and contains 4500, = 4 weeks later
Requirement: W(4) is the number of gallons left 4 weeks later
Equation: W(t) = 4500 - 7t
Solution: W(4) = 4500 - 7(4)
W(4) = 4500 - 28
W(4) = 4472
Answer: There are 4472 gallons left 4 weeks later.
b. How many weeks until the tank is half-full?
Given: - 7 gallons of water, tank is full and contains 4500, the tank is half-full = 0.5
Requirement: t when W(t) = 2250
Equation: W(t) = 0.5 * 4500 = 2250
Solution: W(t) = 0.5 * 4500 = 2250
2250 = 4500 - 7t
2250 - 4500 = - 7t
-2250 = -7t
divide both sides
321.43143 = t
Answer: There will be 321.43143 weeks until the tank is half-full.
c. How many weeks until the tank is empty?

We can procceed

[dusk widget]

alright

empty means W(t) = 0

now it's just the same thing as part b; we gotta solve for t!

[timid silo]

W(t) = 0 * 4500 = 0
0 = 4500 - 7t
0 = 4500 = - 7t
-4500 = -7t
divide both sides
642.857142857 = t

correct?

[dusk widget]

it's all correct except that I'm not sure what the very first line is supposed to mean

[timid silo]

oh sorry

[timid silo]

correct?

[dusk widget]

,calc 4500/7

[warm shale]

Result:

642.85714285714

[dusk widget]

yes, it looks good!

congratulations on Active!

you just got the role, it seems

[timid silo]

A water tank develops a leak. Each week, the tank loses 7 gallons of water due to the
leak. Initially, the tank is full and contains 4500. Using GRESA method.
a. How many gallons left 4 weeks later?
Given: - 7 gallons of water, + tank is full and contains 4500, = 4 weeks later
Requirement: W(4) is the number of gallons left 4 weeks later
Equation: W(t) = 4500 - 7t
Solution: W(4) = 4500 - 7(4)
W(4) = 4500 - 28
W(4) = 4472
Answer: There are 4472 gallons left 4 weeks later.
b. How many weeks until the tank is half-full?
Given: - 7 gallons of water, tank is full and contains 4500, the tank is half-full = 0.5
Requirement: t when W(t) = 2250
Equation: W(t) = 0.5 * 4500 = 2250
Solution: W(t) = 0.5 * 4500 = 2250
2250 = 4500 - 7t
2250 - 4500 = - 7t
-2250 = -7t
divide both sides
321.43143 = t
Answer: There will be 321.43143 weeks until the tank is half-full.
c. How many weeks until the tank is empty?
Given: - 7 gallons of water, tank is full and contains 4500, the tank is empty = 0
Requirement: t when W(t) = 2250
Equation: W(t) = 0 * 4500 = 2250
Solution: W(t) = 0 * 4500 = 0
0 = 4500 - 7t
0 = 4500 = - 7t
-4500 = -7t
divide both sides
642.857142857 = t
643.143 = t
Answer: There will be 643.143 weeks until the tank is empty.

Correct?

[dusk widget]

642.857142857 = t
643.143 = t

eh?

[timid silo]

I round it up

[dusk widget]

you just said that two different numbers equal t >.<

but why keep the decimal then?

[timid silo]

erm cause

Cause

just

[dusk widget]

if you're gonna round, you should drop it

[timid silo]

JUST ITS ACCURATE

[dusk widget]

what

t = 642.857142857 is precisely when the tank becomes empty

not t = 643.143

[timid silo]

oh okay

A water tank develops a leak. Each week, the tank loses 7 gallons of water due to the
leak. Initially, the tank is full and contains 4500. Using GRESA method.
a. How many gallons left 4 weeks later?
Given: - 7 gallons of water, + tank is full and contains 4500, = 4 weeks later
Requirement: W(4) is the number of gallons left 4 weeks later
Equation: W(t) = 4500 - 7t
Solution: W(4) = 4500 - 7(4)
W(4) = 4500 - 28
W(4) = 4472
Answer: There are 4472 gallons left 4 weeks later.
b. How many weeks until the tank is half-full?
Given: - 7 gallons of water, tank is full and contains 4500, the tank is half-full = 0.5
Requirement: t when W(t) = 2250
Equation: W(t) = 0.5 * 4500 = 2250
Solution: W(t) = 0.5 * 4500 = 2250
2250 = 4500 - 7t
2250 - 4500 = - 7t
-2250 = -7t
divide both sides
321.43143 = t
Answer: There will be 321.43143 weeks until the tank is half-full.
c. How many weeks until the tank is empty?
Given: - 7 gallons of water, tank is full and contains 4500, the tank is empty = 0
Requirement: t when W(t) = 2250
Equation: W(t) = 0 * 4500 = 2250
Solution: W(t) = 0 * 4500 = 0
0 = 4500 - 7t
0 = 4500 = - 7t
-4500 = -7t
divide both sides
643 = t
Answer: There will be 643 weeks until the tank is empty.

[timid silo]

Correct?

[dusk widget]

I would probably keep it as a decimal, tbh

[timid silo]

ANO BA TALAGA

ong

[dusk widget]

if your answer for part b was a decimal, your answer for part c should be too

[timid silo]

JUST IT'S FINE

[dusk widget]

for consistency reasons

but if you insist, then yes

this is fine

[timid silo]

😭

[dusk widget]

it's just about consistency, sorry

if both answers were decimals, it's okay

if both are whole numbers, it's okay

but one of each feels... iffy

[timid silo]

t = 643.143 there this is rounded up

A water tank develops a leak. Each week, the tank loses 7 gallons of water due to the
leak. Initially, the tank is full and contains 4500. Using GRESA method.
a. How many gallons left 4 weeks later?
Given: - 7 gallons of water, + tank is full and contains 4500, = 4 weeks later
Requirement: W(4) is the number of gallons left 4 weeks later
Equation: W(t) = 4500 - 7t
Solution: W(4) = 4500 - 7(4)
W(4) = 4500 - 28
W(4) = 4472
Answer: There are 4472 gallons left 4 weeks later.
b. How many weeks until the tank is half-full?
Given: - 7 gallons of water, tank is full and contains 4500, the tank is half-full = 0.5
Requirement: t when W(t) = 2250
Equation: W(t) = 0.5 * 4500 = 2250
Solution: W(t) = 0.5 * 4500 = 2250
2250 = 4500 - 7t
2250 - 4500 = - 7t
-2250 = -7t
divide both sides
321.43143 = t
Answer: There will be 321.43143 weeks until the tank is half-full.
c. How many weeks until the tank is empty?
Given: - 7 gallons of water, tank is full and contains 4500, the tank is empty = 0
Requirement: t when W(t) = 2250
Equation: W(t) = 0 * 4500 = 0
Solution: W(t) = 0 * 4500 = 0
0 = 4500 - 7t
0 = 4500 = - 7t
-4500 = -7t
divide both sides
642.857142857 = t
Answer: There will be 642.857142857 weeks until the tank is empty.

[dusk widget]

why are you rounding up?

[dusk widget]

.

[timid silo]

BECAUSE MRSJGGEOIJOUOGBGOI

[timid silo]

Correct?

[dusk widget]

seems good to me

congratulations!

[timid silo]

Okie

[dusk widget]

you finished this problem

pat yourself on the back

[timid silo]

BWAHAHHAHAH

Okay lemme just write it then were done

[dusk widget]

[obtuse pebble]

@timid silo Has your question been resolved?

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[analog rivet]

how would one use the sine ratio to solve this?

the marking schemes suggests using a "sine ratio"

not the cosine rule

i do not comprehend how this would work

thanks :)

[obtuse pebble]

@analog rivet Has your question been resolved?

[analog rivet]

<@&286206848099549185>

[obtuse pebble]

@analog rivet Has your question been resolved?

[analog lynx]

do you know how to solve something like this?

[obtuse pebble]

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Use .reopen if this was a mistake.

[obtuse pebble]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[manic moon]

Is there a general algorithm that isnt numerical for finding x in a sum from i=0 to x of f(i) that equals K, when K and f(x) are known?

[kind hawk]

we cant even solve polynomials of degree >=5

your problem could be much more general

or at least it feels that way. maybe it isnt. having the restriction that x is a natural number could make it easier

but in general you cant even hope for a closed form of such a sum