Find the rate of change of the volume of a sp | vedclass

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(A) The volume $V$ of a sphere is given by $V = \frac{4}{3} \pi r^3$ and the surface area $A$ is given by $A = 4 \pi r^2$.To find the rate of change o

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(A) The volume $V$ of a sphere is given by $V = \frac{4}{3} \pi r^3$ and the surface area $A$ is given by $A = 4 \pi r^2$.To find the rate of change of volume with respect to surface area,we calculate $\frac{dV}{dA} = \frac{dV/dr}{dA/dr}$.Differentiating $V$ with respect to $r$: $\frac{dV}{dr} = 4 \pi r^2$.Differentiating $A$ with respect to $r$: $\frac{dA}{dr} = 8 \pi r$.Therefore,$\frac{dV}{dA} = \frac{4 \pi r^2}{8 \pi r} = \frac{r}{2}$.At $r = 2 \ cm$,the rate of change is $\frac{dV}{dA} = \frac{2}{2} = 1 \ cm$.

Find the rate of change of the volume of a sphere with respect to its surface area when the radius is $2 \ cm$.

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