<s>[INST] Find dV/dA in terms of r. The volume of a sphere, $V$ cm³, of radius $r$ is given by the formula $V = \frac{4}{3} \pi r^3$. The surface area of a sphere $A$ cm² of radius $r$ cm is given by the formula $A=4\pi r^2$. Find $dV/dA$ in terms of $r$.
Here's my workings to the question:
$$V= \frac{\frac{4}{3}\pi r^{3}}{4\pi r^2}A = \frac{1}{3}rA$$
So, $$\frac{dV}{dA} = \frac{1}{3}r =\frac{r}{3}.$$
I am not sure about this answer, so it would help to know if anyone got the same answer. Thank you!
 [/INST] Given $V= \frac{4\pi}3 r^3$ and $A=4\pi r^2$, we have
$$V= \frac{4\pi}3\left(\frac{A}{4\pi}\right)^{3/2}= \frac{A^{3/2}}{3\sqrt{4\pi}}
$$
Then,
$$\frac{dV}{dA}= \frac12\frac{\sqrt A}{\sqrt{4\pi}}= \frac r2
$$
</s>