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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