Question

You work at a canning factory that's producing cans for a new brand of soup. You need to decide what size the cans should be. The soup cans can have a radius of either 2 in, 2.5 in, 3 in, or 3.5 in. The cans need to hold a volume of exactly 90 in3. The company wants the cans to be no more than 5 inches tall, and it wants the cans to have the greatest lateral surface area possible so it can print more information on the side of the cans. To solve this problem, you will fill in this table with the surface area and volume of each cylinder:First, calculate the height each can must be, given the radius and volume.
b. Now calculate the lateral surface area for each possible can.
















c. Based on the requirements for the can, which can should you make?

Answer 1

The volume of a cylinder is V = pi*(r^2)*h and the lateral surface area is S = 2pi*r*h. 

In this case we must have V = pi*(r^2)*h = 90, so h = 90 / (pi*r^2). 

Thus S = 2pi*r*(90 / (pi*r^2)) = 180 / r. So since S is inversely proportional to r we 

will want the smallest possible radius for the cans as possible. 

Now with r = 2 we require h = 90 / (pi*r^2) = 90 / (4*pi) = 7.162 inches, but since 

this is greater than 5 inches we can't use this can radius. 

With r = 2.5 inches we have h = 90 / (pi*(2.5)^2) = 4.584 inches to 3 decimal places, 

resulting in a lateral surface area of S = 180 / 2.5 = 72 square inches.