# Rate of change of radius and volume

They put a gas bubble in someone's eye. The volume of a gas bubble changes from $0.4$ $cc$ to $1.6$ $cc$ in $74$ hours. Assuming that the rate of change of the radius is constant, find

• (a) The rate at which the radius changes;
• (b) The rate at which the volume of the bubble is increasing at any volume $V$;
• (c) The rate at which the volume is increasing when the volume

is $1$ $cc$. (Note: The volume of a ball of radius $r$ is $\frac{4}{3}\pi r^3$. Assume the bubble is spherical.)

Explanation would be appreciated.

I did differentiate the $\frac{4}{3}\pi r^3$ with respect to r that is $4\pi r^2$ and then made that equal to $1.66-06$ which is the rate change of $V$. But I Don't know if I am doing it right.

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Remember that if the radius grows by a factor of x then the volume grows by X^3. So if the volume grew 4 times then the radius grew by... – DannyDan Nov 7 '13 at 18:10

$v=\frac{4\pi r^3}{3}$

or

$r=(3v/4\pi)^{1/3}$

$\frac{dv}{dt}=4\pi r^2\frac{dr}{dt}$ substitute for $r$ we get

and letting $dr/dt=K$ (constant) then

$v^{-2/3}dv=4\pi K(3/(4\pi))^{2/3} dt$

$3v^{1/3} = 4\pi K(3/(4\pi))^{2/3} dt$

Integrating between limits $0.4$ and $1.6$ for V and $t=0$ to $74$, we get $K= \frac{1.6^{1/3}-.4^{1/3}}{74 (4\pi/3)^{1/3}}$ which works out to $3.628E-3 cm/hr$ as the rate of change of radius.

b) $dv/dt=(36\pi)^{1/3} KV^{2/3}$

c) substitute $v=1$ in above we get $(36\pi)^{1/3} K$

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It would be alot easier for part 1 to just: $$K = \frac{\sqrt[3]{1.6*3/4/\pi}-\sqrt[3]{0.4*3/4/\pi}}{74} cm/hr$$ but they both get the same answer – kaine Nov 7 '13 at 19:12

Hint: what are the starting and ending radii? Let $t$ be the number of hours from the start. Now write an equation $r=$ some function of $t$. Use that to write another equation $V=$ some function of $t$ Now use these functions to answer the questions.

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If the radius increases at a constant rate, $r = \alpha t+r_o$ for some constant $\alpha$.

this simplifies the questions to:

a) What are the values of $\alpha$ and $r_o$? (should be solvable from the data)

b) Write $V(t)$, solve for $\frac{dV}{dt}$, and write that in terms of just V.

c) Specifically plug $V=1 cc$ into the answer for b.

Note that we are hesitating to give a direct answer as this appears to be homework but I will gladly confirm any answers you come up with.

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