Solve the problem.
-A balloon used in surgical procedures is cylindrical in shape. As it expands outward, assume that the length remains a constant $40.0 \mathrm {~mm}$. Find the rate of change of surface area with respect to radius when the radius is $0.080 \mathrm {~mm}$. (Answer can be left in terms of $\pi$ ).
A) $80.0 \pi \mathrm { mm } ^ { 2 } / \mathrm { mm }$
B) $40.32 \pi \mathrm { mm } ^ { 2 } / \mathrm { mm }$
C) $80.32 \pi \mathrm { mm } ^ { 2 } / \mathrm { mm }$
D) $40.16 \pi \mathrm { mm } ^ { 2 } / \mathrm { mm }$

Question

Quizplus · Accepted Answer

The surface area $S$ of a cylinder is given by $S = 2\pi r^2 + 2\pi rh$, where $r$ is the radius and $h$ is the height. Differentiating $S$ with respect to $r$ gives $\frac{dS}{dr} = 4\pi r + 2\pi h$. Substituting $r = 0.080 \, \mathrm{mm}$ and $h = 40.0 \, \mathrm{mm}$ into this formula yields $\frac{dS}{dr} = 4\pi(0.080) + 2\pi(40.0) = 0.32\pi + 80\pi = 80.32\pi \, \mathrm{mm}^2/\mathrm{mm}$.

Solve the Problem 40.0 mm40.0 \mathrm {~mm}40.0 mm Find the Rate of Change of Surface Area with Cylindrical

Solve the Problem $40.0 \mathrm {~mm}$ Find the Rate of Change of Surface Area with Cylindrical