# Rates of change

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#1
The volume of a spherical bubble is increasing at a constant rate. Show that the rate of increase of the radius r of the bubble is inversely proportional to r^2. Volume of a sphere is 4/3 pi r^3

I’ve found that dv/dr=4pi r^2, but I’m not sure how to form a rates equation or show how the rate of increase is inversely proportional. Any help would be greatly appreciated )
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1 month ago
#2
(Original post by student38910)
The volume of a spherical bubble is increasing at a constant rate. Show that the rate of increase of the radius r of the bubble is inversely proportional to r^2. Volume of a sphere is 4/3 pi r^3

I’ve found that dv/dr=4pi r^2, but I’m not sure how to form a rates equation or show how the rate of increase is inversely proportional. Any help would be greatly appreciated )
What differential equation do you get from the first sentance?
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#3
(Original post by mqb2766)
What differential equation do you get from the first sentance?
4 x pi x r^2
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1 month ago
#4
(Original post by student38910)
4 x pi x r^2
No. What are the *s in
d*/d* = *
for the first sentance?
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#5
(Original post by mqb2766)
No. What are the *s in
d*/d* = *
for the first sentance?
Dr/dv?
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1 month ago
#6
(Original post by student38910)
Dr/dv?
If you said that your money (in a bank) was increasing at a constant rate, how would you write that as a differential equation.
Then apply the same logic here.
Last edited by mqb2766; 1 month ago
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#7
(Original post by mqb2766)
If you said that your money (in a bank) was increasing at a constant rate, how would you write that as a differential equation.
Then apply the same logic here.
I honestly don’t know
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1 month ago
#8
(Original post by student38910)
I honestly don’t know
Why not have a think for a bit? Think/read about simple interest (and the difference from compound interest).

If you talk about the rate of increase of something, what does that tell you about the differential equation? If you don't like the money example, how is displacement increasing at a constant rate (velocity), or virus infections are increasing at a constant rate or ...
Last edited by mqb2766; 1 month ago
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#9
(Original post by mqb2766)
Why not have a think for a bit? Think/read about simple interest (and the difference from compound interest).

If you talk about the rate of increase of something, what does that tell you about the differential equation? If you don't like the money example, how is displacement increasing at a constant rate (velocity), or virus infections are increasing at a constant rate or ...
the increase of something means that the dy/dx>0?
the increase of interest is the amount x no of years x the interest rate.

But what does this have to do with showing the that the rate of increase is inversely proportional to r2
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1 month ago
#10
(Original post by tej3141)
i...
pls read the sticky at the top of the maths forum about providing hints rather than solutions. Thanks. Edit / delete the post?
Last edited by mqb2766; 1 month ago
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1 month ago
#11
sorry i didnt know ill delete it now

(Original post by mqb2766)
pls read the sticky at the top of the maths forum about providing hints rather than solutions. Thanks. Edit / delete the post?
1
1 month ago
#12
(Original post by student38910)
the increase of something means that the dy/dx>0?
the increase of interest is the amount x no of years x the interest rate.

But what does this have to do with showing the that the rate of increase is inversely proportional to r2
You may have been slightly side-tracked by the discussion about interest rates - I think the point was to get you thinking about "rates of change" in general.

You actually have most of the pieces that you need already! Unless otherwise specified, when we talk about the rate at which something is changing we usually mean the rate over time. The rate of change of a quantity Q with respect to time is just the first derivative dQ/dt.

In this case we are considering volume V of a spherical bubble and we are told in the question that this volume increases at a constant rate i.e. dV/dt = k, where k is some constant.

Now, the question wants us to make a statement about the rate of increase of the radius r, that is, we are trying to write down an equation that starts dr/dt = ???. We are hoping that those ???s can be replaced by a formula that corresponds to an inverse square relationship. You should know from GCSE what it means for one thing to be inversely proportional to the square of another.

The "missing piece" of the jigsaw is a rule that connects dV/dt with dr/dt, given that you are told that V = (4/3)(pi)r^3.

Do you know how to proceed now? If I said "chain rule for differentiation" does this ring any bells?
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