A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 30 cubic centimeters. Find the radius, r, of the cylinder that Produces the minimum surface area. Round your answer to two decimal places.
A) $r \approx 2.68 \mathrm {~cm}$
B) $r \approx 2.39 \mathrm {~cm}$
C) $r \approx 1.93 \mathrm {~cm}$
D) $r \approx 3.57 \mathrm {~cm}$
E) $r \approx 2.52 \mathrm {~cm}$

Question

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 30 cubic centimeters. Find the radius, r, of the cylinder that Produces the minimum surface area. Round your answer to two decimal places. 
A) $r \approx 2.68 \mathrm {~cm}$
B) $r \approx 2.39 \mathrm {~cm}$
C) $r \approx 1.93 \mathrm {~cm}$
D) $r \approx 3.57 \mathrm {~cm}$
E) $r \approx 2.52 \mathrm {~cm}$

Quizplus · Accepted Answer

To minimize the surface area, we set up the volume equation $ V = \frac{4}{3}\pi r^3 + \pi r^2 h = 30 $ and solve for $ h $ in terms of $ r $. Then, we express the surface area $ S = 2\pi r^2 + 2\pi rh $ in terms of $ r $ only. By differentiating $ S $ with respect to $ r $ and setting the derivative to zero, we find the critical points. Evaluating these points gives the minimum surface area at $ r \approx 1.93 \mathrm{~cm} $.

A Solid Is Formed by Adjoining Two Hemispheres to the Ends