I am having problems solving the following question.

The volume, $V$, of a sphere of radius r is given by $V=f(r)=\frac{4}{3}\pi r^3$. Calculate the instantaneous rate of change of the volume, $V$, with the respect to change of the radius, $r$, at $r=36.4$.

I assume the answer to this question would be $f\prime(36.4)$

where $f\prime$ is equal to;

$f\prime(x) = 4\pi x^2 \\ f\prime(36.4) = 4\pi (36.4)^2 \\= 16649.93$

Although this is not the solution. Please advise me where I have went wrong.

  • $\begingroup$ Sorry, my appologies, I left a typo while formatting my question. I meant to put $\frac{4}{3}$ and not $\frac{3}{4}$ $\endgroup$ – Miroslav Glamuzina Oct 12 '15 at 23:03
  • $\begingroup$ Then you were doing right! $\endgroup$ – Megadeth Oct 12 '15 at 23:04
  • $\begingroup$ That is what I believed, but the solution I have is apparently incorrect? I just wanted to confirm my answer. Thankyou. $\endgroup$ – Miroslav Glamuzina Oct 12 '15 at 23:05
  • $\begingroup$ Looks good to me. You've been asked to compute for $\frac{dV(r)}{dr}$ for some value of $r$ and that's what you did. $\endgroup$ – mopy Oct 12 '15 at 23:08
  • $\begingroup$ Closer to $16649.94$ (kidding). Maybe they wanted you to round. $\endgroup$ – André Nicolas Oct 12 '15 at 23:11

This is a related rates problem, and it seems your trouble is in the formula you're using. The volume of a sphere is $V = \frac{4}{3}\pi r^3$, not $\frac{3}{4}$.

Other than that, you have the right idea.

EDIT: Just saw that you fixed that.

  • $\begingroup$ What is the rate of change of the radius? That's a key piece of information that is missing. $\endgroup$ – Cody Rudisill Oct 12 '15 at 23:09
  • $\begingroup$ The right side of your formula should be expressed 4*pi*(36.4)^2*(dr/dt), where dr/dt is the rate of change of the radius. $\endgroup$ – Cody Rudisill Oct 12 '15 at 23:11

$V(r)=\frac{4}{3}\pi r^3$. Using the chain rule, we have $\frac{dV}{dr}=4\pi r^2\cdot\frac{dr}{dt}$. We know $r=36.4$. However, the radius depends on time and you did not provide that info. In your problem, they'll give you $r(t)$, and take it's derivative. Then your answer is $\frac{dV}{dr}=4\pi\cdot 36.4^2\cdot\frac{dr}{dt}$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.