Question

As a preparation for the long bright summer days, Dr. Acula plans to store gourmet plasma in closed tin cans that have the shape of a cylinder with volume V. As an environmentally conscious member of the community, he wants to use as little metal as possible.

A) What is the height h(r) for the can with minimum surface area in terms of the radius r of the bottom?
B) What is the minimum surface area for the can in terms of the portion size V?

Answer 1

To find the height h(r) for the can with minimum surface area, take the derivative of the surface area function and set it equal to zero. The height for the can with minimum surface area is equal to the radius of the bottom of the can. The minimum surface area is given by 4πr² in terms of the portion size V.

Step-by-step explanation:

In order to minimize the surface area of a cylinder, we need to find the height that corresponds to the minimum surface area. Let's assume the radius of the bottom of the cylinder is r and the height is h.

a. To find the height h(r) for the can with minimum surface area, we need to minimize the surface area function. The surface area of a cylinder is given by:

A = 2πr² + 2πrh

To minimize the surface area, we can take the derivative of the surface area function with respect to h and set it equal to zero:

dA/dh = 2πr + 2πr(1 - h/r) = 2πr(2 - h/r) = 0

Solving for h, we find h(r) = r. Therefore, the height for the can with minimum surface area is equal to the radius of the bottom of the can.

b. The minimum surface area for the can in terms of the portion size V can be obtained by substituting the height h(r) into the surface area function:

A = 2πr² + 2πr(r) = 4πr²

The minimum surface area is given by 4πr² in terms of the portion size V.