What will be the rate of change of the volume | vedclass

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(D) Let the radius of the sphere be $r$ and its diameter be $D = 2r$. The volume $V$ of the sphere is given by $V = \frac{4}{3} \pi r^3$. We need to f

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$2 \pi r^2$

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(D) Let the radius of the sphere be $r$ and its diameter be $D = 2r$. The volume $V$ of the sphere is given by $V = \frac{4}{3} \pi r^3$. We need to find the rate of change of volume with respect to the diameter,which is $\frac{dV}{dD}$. Since $D = 2r$,we have $r = \frac{D}{2}$. Substituting $r$ in the volume formula: $V = \frac{4}{3} \pi (\frac{D}{2})^3 = \frac{4}{3} \pi (\frac{D^3}{8}) = \frac{1}{6} \pi D^3$. Now,differentiate $V$ with respect to $D$: $\frac{dV}{dD} = \frac{d}{dD} (\frac{1}{6} \pi D^3) = \frac{1}{6} \pi (3D^2) = \frac{1}{2} \pi D^2$. Substituting $D = 2r$ back into the expression: $\frac{dV}{dD} = \frac{1}{2} \pi (2r)^2 = \frac{1}{2} \pi (4r^2) = 2 \pi r^2$. Thus,the correct option is $D$.

What will be the rate of change of the volume of a sphere with radius $r$,with respect to its diameter?

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