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- #1

#### wolfsprint

##### New member

- Dec 31, 2012

- 17

- Thread starter wolfsprint
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- #1

- Dec 31, 2012

- 17

- Jan 30, 2012

- 2,502

Express the volume and the surface area through r(t) and differentiate them.

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- #3

- Dec 31, 2012

- 17

I tried and I couldn't, can you please show the work?

- Jan 30, 2012

- 2,502

Well, start by showing your work in expressing the volume and the surface area through the radius. You should not do it yourself; you should just find the formulas in a textbook or online. Then what exactly is your difficulty in differentiating? You need to use the chain rule because the volume is a function of r and r is a function of t.I tried and I couldn't, can you please show the work?

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- #5

- Dec 31, 2012

- 17

So if we say that the surface area = x ,then

V=1/3 r . X

But then when i differentiate it it turns out like this ,

Dv/dt = 1/3 dr/dt . Dx/dt , and that prove needs dv/dt = 1/2 r . Dx/dt

- Jan 30, 2012

- 2,502

Since you found the relationship between v and x, you must know that $x = 4\pi r^2$, so this is the second equation that you have. You can also find this in Wikipedia.The only equation i managed to find is v=4/3 "pi" r^3 which is he volume of the sphere.

So if we say that the surface area = x ,then

V=1/3 r . X

In post #2, I recommended expressing both v and x only through r and then differentiating them as compositions v(r(t)) and x(r(t)). Here you expressed v through xBut then when i differentiate it it turns out like this ,

Dv/dt = 1/3 dr/dt . Dx/dt , and that prove needs dv/dt = 1/2 r . Dx/dt

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- #7

- Dec 31, 2012

- 17

I still dont understand , what do you by "expressing"?

- Jan 30, 2012

- 2,502

By expressing $v$ and $x$ through $r$ I mean finding formulas containing constants and $r$ only that give the values of $v$ and $x$. These formulas are $v=\frac{4}{3}\pi r^3$ and $x=4\pi r^2$. Then we have $\frac{dv}{dt}=4\pi r^2\frac{dr}{dt}$. Similarly, find $\frac{dx}{dt}$ (keep in mind that $r$ is a function of $t$, i.e., $x(t)$ is a composite function $x(r(t))$, just like $v(r(t))$) and compare the results.I still dont understand , what do you by "expressing"?

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- #9

- Dec 31, 2012

- 17

- Jan 30, 2012

- 2,502

\[\frac{dv}{dt}=4\pi r^2\frac{dr}{dt}\tag1\]

Next, since $x=4\pi r^2$, we have

\[\frac{dx}{dt}=8\pi r\frac{dr}{dt}\tag2\]

Comparing the right-hand sides of (1) and (2), we see that $\frac{dv}{dt}=\frac{1}{2}r\frac{dx}{dt}$.

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- #11

The volume of the sphere is:

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

The surface area is:

(2) $\displaystyle x=4\pi r^2$

We are asked to show:

$\displaystyle \frac{dv}{dt}=\frac{1}{2}r\frac{dx}{dt}$

Now, differentiating (1) and (2) with respect to time

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- #12

- Dec 31, 2012

- 17

Thanks alot! i cant believe i missed that