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Question

Find the rate of change of the volume of a sphere with respect to its surface area when the radius is varying.


Solution

We know that $$Surface\ area=2\pi r^2+2\pi rh$$
$$Volume\  v=\dfrac{4}{3}\pi r^3$$
$$\dfrac{\mathrm{d} s}{\mathrm{d} r}=2\pi(2r+h)$$
$$\dfrac{\mathrm{d}v }{\mathrm{d} r}=4\pi r^2$$
$$\dfrac{\mathrm{d} v}{\mathrm{d} s}=\dfrac{4\pi r^2}{2\pi(2r+h)}$$
$$=>\dfrac{\mathrm{d} v}{\mathrm{d} s}=\dfrac{2r^2}{(2r+h)}$$

Mathematics

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