Surface Area and Volume Question? (Please help!!!!)

Question

Sandra's Soup Shop is a very popular lunch spot in the downtown area. After a great deal of planning, Sandra decides to sell two of the most popular types of soup in aluminum cans. She will first sell them at her restaurant, and if things go well she would like to sell the soup in grocery stores. 

 

Sandra wants each can to hold 355 mL of soup. Since 1 mL=1 cubic centimeter, this means that the volume of a can must be 355 cm^3. After Sandra has determined the dimensions of the can, an artist will design a rectangular label that will be affixed to the lateral surface of the can. 

 

Determine the dimensions of the can that will minimize the amount of aluminum needed. Then calculate the dimensions of the rectangular label. Round your answers to the nearest hundredth of a centimeter. 

Answer 1

The solution to this question is more complex than it appears.

Part 1 the can
From the chart of standard can sizes, we find that a #1 tall can will hold 12.0 ounces, which is equal to 355cc.

The dimensions of such cans are a base of diameter 2 11/16”, and a height of 4 13/16”.

When cans are made of aluminum, the base is formed with the cylinder, so no extra part is needed. But, each can will need a lid, whose diameter must be slightly larger than the diameter of the base (usually 3/8"). So the diameter of the lid will be 2 11/16” + 3/8” or 3 1/16”.

We will need the area of the base to figure the total area. Since there will be no overlap on the base, its area will be pi times the square of the radius.

Then, we need to calculate the area of the cylinder, which will be given by the perimeter of the base times the height. The perimeter of the base is pi x diameter.

So, the surface area of the can will be the area of the cylinder plus the area of the base plus the area of the lid with its overlap.

Part 2 the label
The label will be a rectangle slightly longer than perimeter of the base to allow for an overlap which is needed so that the label can be glued together. Usually, that extra length is 3/8". So, its length will be the perimeter of the base plus 3/8”.

The label will have a height which will be slightly less than the height of the can, because labels never cover the fold where the lid is sealed to the cylinder. That distance will be half of the lid overlap or 3/16”. So, the height of the label will be 4 5/8”.

From this information you should be able to calculate the surface area of the can, and the size of the label.