The rate of change of the volume of a sphere | vedclass

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Question

(B) The volume $V$ of a sphere with radius $r$ is given by the formula: $V = \frac{4}{3} \pi r^3$. To find the rate of change of the volume with respe

Vedclass · Accepted Answer

$196 \pi \ cm^2$

Vedclass · Answer

(B) The volume $V$ of a sphere with radius $r$ is given by the formula: $V = \frac{4}{3} \pi r^3$. To find the rate of change of the volume with respect to its radius,we differentiate $V$ with respect to $r$: $\frac{dV}{dr} = \frac{d}{dr} (\frac{4}{3} \pi r^3) = \frac{4}{3} \pi (3r^2) = 4 \pi r^2$. Now,we evaluate this derivative at $r = 7 \ cm$: $\frac{dV}{dr} \Big|_{r=7} = 4 \pi (7)^2 = 4 \pi (49) = 196 \pi$. The unit of the rate of change of volume with respect to radius is $\frac{cm^3}{cm} = cm^2$. Therefore,the rate of change is $196 \pi \ cm^2$.

The rate of change of the volume of a sphere with respect to its radius when $r = 7 \ cm$ is . . . . . . .

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