Rates of change exam question

Sophie02

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Hi,

I was wondering how to do this question? Everything I seem to try, the result never seems to be the same as the equation that you need to prove? And then for part b, I don’t know what bits of data I would be using to go where?

thankyou
 

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lex

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Do you want to post an example of what you are doing for (a) and we can see what is happening? It may simply be a matter of getting your work to 'look like' their answer.
 

HallsofIvy

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In order to get that equation, the balloon is a sphere and the volume of a sphere of radius r is \(\displaystyle V= (4/3)\pi r^3\). Then \(\displaystyle dV/dr= 4\pi r^2\) and \(\displaystyle dV/dt= 4\pi r^2 dr/dt\). If that is a constant, C, the \(\displaystyle 4\pi r^2 dr/dt= C\) so that \(\displaystyle dr/dt=C/4\pi r^2= k/r^2\) with \(\displaystyle k= C/4\pi\) a constant.
 

lex

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...so that \(\displaystyle dr/dt=C/4\pi r^2= k/r^2\) with \(\displaystyle k= C/4\pi\) a constant.
\(\displaystyle C\) is negative, but \(\displaystyle k\) is positive. So \(\displaystyle k= \boldsymbol{-C}/4\pi \hspace3ex\)( and \(\displaystyle \tfrac{dr}{dt}=-\tfrac{k}{r^2}\))
 
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Sophie02

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In order to get that equation, the balloon is a sphere and the volume of a sphere of radius r is \(\displaystyle V= (4/3)\pi r^3\). Then \(\displaystyle dV/dr= 4\pi r^2\) and \(\displaystyle dV/dt= 4\pi r^2 dr/dt\). If that is a constant, C, the \(\displaystyle 4\pi r^2 dr/dt= C\) so that \(\displaystyle dr/dt=C/4\pi r^2= k/r^2\) with \(\displaystyle k= C/4\pi\) a constant.
Ohh that makes more sense now because i was getting 4pi and I didn’t know how to get rid of it. But it’s just because it’s a constant included within k
 

Sophie02

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How would you go about doing part b?
 

Subhotosh Khan

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How would you go about doing part b?
What have YOU tried for this EXAM problem?

Are you allowed to ask for external assistance?
 

lex

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Ohh that makes more sense now because i was getting 4pi and I didn’t know how to get rid of it. But it’s just because it’s a constant included within k
Do you want to post an example of what you are doing for (a) and we can see what is happening? It may simply be a matter of getting your work to 'look like' their answer.
That's what I suspected and why I offered to look at your work if you posted it.
 

Sophie02

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What have YOU tried for this EXAM problem?

Are you allowed to ask for external assistance?
Yes I’m allowed external help. I’m just doing practise papers for revision!
 

Subhotosh Khan

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Yes I’m allowed external help. I’m just doing practise papers for revision!
OK

So where is your work/thoughts regarding this problem?
 

Sophie02

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but as you can see I don’t have an equation so I’ve definitely don it wrong
 

Subhotosh Khan

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but as you can see I don’t have an equation so I’ve definitely don it wrong
V= (4/3)*π*r^3 → dV/dr = 4 * π * r^2

so the DE is:

(-80*π) = 4 * π * r^2 * dr/dt

There is your equation (DE) ...... continue......
 

Subhotosh Khan

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is this correct?
Should the 'r' (radius) increase or decrease as 't' (time) increases?

Do you get r = 20 at t = 5 from the equation you have derived?

and

Do you get r = 40 at t = 0 from the equation you have derived?
 
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