Rate of change (ice ball melting)

[jisbon]

Homework Statement

Assume a ice ball melts so that volume decreases proportionately to its surface area. It takes eight hours to melt down to 1/8 of its original volume. How much time does it take to melt completely?

Relevant Equations

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So to begin this question, I do know that volume =4/3 pi r cubed, while the surface area) 4 pi r squared.
I will like to clarify some things about the question:
1) does the first sentence means dv/dt is proportional to 4 pi r squared?
2) given the second sentence how am I able to construct an equation out of the values given to me?

Thanks

 

[Filip Larsen]

1) If you write v for volume then yes.
2) If you assume a certain proportionality factor (say, k) can you then write up a relationship between change in volume as function of change in time?

 

[Delta2]

My hint will be to consider the radius of the ball $r$ as a function of time $r(t)$. Express volume as function of $r(t)$ then calculate $\frac{dV}{dt}$ with implicit differentiation, and setup the differential equation $$\frac{dV}{dt}=k4\pi r^2(t)$$
From what I can infer that differential equation which will have unknown the function $r(t)$ is pretty easy. Use then the information from the second sentence to find a relationship between $r(8)$ and $r(0)$ and then also using the solution of the diff. equation you should be able to determine the constant k. Then knowing k all you need to do is go back to the solution of differential equation and solve the algebraic equation$r(t_0)=0$ that is to find the time $t_0$ for which the radius of the iceball becomes zero.

 

[jisbon]

My hint will be to consider the radius of the ball $r$ as a function of time $r(t)$. Express volume as function of $r(t)$ then calculate $\frac{dV}{dt}$ with implicit differentiation, and setup the differential equation $$\frac{dV}{dt}=k4\pi r^2(t)$$
From what I can infer that differential equation which will have unknown the function $r(t)$ is pretty easy. Use then the information from the second sentence to find a relationship between $r(8)$ and $r(0)$ and then also using the solution of the diff. equation you should be able to determine the constant k. Then knowing k all you need to do is go back to the solution of differential equation and solve the algebraic equation$r(t_0)=0$ that is to find the time $t_0$ for which the radius of the iceball becomes zero.

Ok, from what I gathered.
$\frac{dv}{dt} = kA$
$\frac{dv}{dt} = 4\pi r^2\frac{dr}{dt}$
$4\pi r^2\frac{dr}{dt} = kA$
$4\pi r^2\frac{dr}{dt} =k (4\pi r^2)$
$\frac{dr}{dt} = k$ (constant)

$V=\frac{4}{3}\pi R^3$
$V'=\frac{1}{8}V$
Let r be radius after 8 hours and R be original radius.
$\frac{4}{3}\pi r^3 =\frac{1}{8}*\frac{4}{3}\pi R^3$
$R= 2r$

So, since the radius decreased wrt time is constant, can I assume since it lost 0.5 of its original radius in 8 hours, is it true that it will take 16 hours for it to melt completely (aka lose all radius)?

 

[Delta2]

I think that is correct.

 

[Filip Larsen]

There may also be a shortcut to take, if one considers how radius will vary over time as the ball melts.

Edit: sorry, missed that the OP actually did derive that dr/dt is constant