Question textThe volume of a circular cylinder is given by the function $V(r) =\pi r^{2}h$, with the radius $r$ and height $h$. For a fixed height, find the rate of change in the volume when the radius is $4$ cm and the height is $9$ cm. (Hints: Find the rate of change in the volume of the cylinder with respect to $r$) Answer 1 Question 1[input] cm $^{3}$/cm

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BlackTom AI · Accepted Answer

The problem asks for the rate of change of the volume with respect to the radius r for a cylinder with fixed height h. Start from the volume formula V(r) = π r^2 h, where h is constant. Differentiate V with respect to r: dV/dr = d/dr (π r^2 h) = 2π r h, since π and h are constants with respect to r.

Next, substitute the given values: r = 4 cm and h = 9 cm. This yields dV/dr = 2π (4) (9) = 72π.

The units come from differentiating volume (cm^3) with respect to radius (cm), giving cm^3 per cm, i.e., cm^3/cm. Therefore, the rate of change in the volume with respect to r at r = 4 cm and h = 9 cm is 72π cm^3/cm.

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