# Maths: Rates of change

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#1
I've been going over and over the following question for hours when it should only take a few minutes, I have been close a few times but not quite getting the end result. Was wondering if anyone could come up with the worked solution.

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.

Many thanks.
0
7 months ago
#2
So you need to start off by coming up with a differential equation for the rate of change of volume, which you know is constant.

Think about what the volume of a sphere is and how you could get from the rate of change of volume to the rate of change of the radius.
1
#3
(Original post by chapmase)
So you need to start off by coming up with a differential equation for the rate of change of volume, which you know is constant.

Think about what the volume of a sphere is and how you could get from the rate of change of volume to the rate of change of the radius.
From differentiating the volume of the sphere, I came up with 4πr^2.
However, the problem I am having is coming up with the rate of changes equation of which I would need to show the inverse proportionality.
0
7 months ago
#4
chain rule on dV/dt
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7 months ago
#5
You are told that the volume of the sphere is increasing at a constant rate, so you can write this as (where c is a constant).

You are asked to show (and therefore find) the rate of increase of the radius, which is

To find this, use the chain rule:

You already have and so need to work out

HINT:

Now you should be able to figure out

HINT: Remember to gather all the constants into one constant, which you can now label k.

You should get
1
#6
(Original post by mathstutor24)
You are told that the volume of the sphere is increasing at a constant rate, so you can write this as (where c is a constant).

You are asked to show (and therefore find) the rate of increase of the radius, which is

To find this, use the chain rule:

You already have and so need to work out

HINT:

Now you should be able to figure out

HINT: Remember to gather all the constants into one constant, which you can now label k.

You should get
That's ever so helpful thankyou, that makes the problem look so much easier to go about and solve.
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7 months ago
#7
No problem! Happy to help.

Please ignore if I'm asking about something you already know, but just in case - are you happy with how to set up differential equations involving a rate of change and a proportion relation?
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#8
(Original post by mathstutor24)
No problem! Happy to help.

Please ignore if I'm asking about something you already know, but just in case - are you happy with how to set up differential equations involving a rate of change and a proportion relation?
Yes thanks, I think the issue was the constants and putting all the constants under one constant, k.
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