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Question

(A) 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 at which the volume is increasing

Vedclass · Accepted Answer

$400 \pi 	ext{ cm}^3/	ext{cm}$

Vedclass · Answer

(A) 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 at which the volume is increasing with respect to the radius,we calculate the derivative $\frac{dV}{dr}$: $\frac{dV}{dr} = \frac{d}{dr} \left( \frac{4}{3} \pi r^3 ight) = \frac{4}{3} \pi (3r^2) = 4 \pi r^2$. We are asked to find this rate when the radius $r = 10 	ext{ cm}$. Substituting $r = 10$ into the derivative: $\frac{dV}{dr} = 4 \pi (10)^2 = 4 \pi (100) = 400 \pi$. Thus,the rate of change of volume with respect to the radius is $400 \pi 	ext{ cm}^3/	ext{cm}$.

$A$ balloon,which always remains spherical,has a variable radius. Find the rate at which its volume is increasing with respect to its radius when the radius is $10 \text{ cm}$.

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