suppose you work for a company that manufactures cylindrical cans. which will cost more to manufacture, a can with a radius of 4 inches and a height of 3 inches, or a can with a radius of 3 inches and a height of 4 inches? assume that the cost of material for the tops and bottoms is $0.80 per square inch and the cost of material for curved surfaces is $0.40 per square inch.
To determine which can will cost more to manufacture, we need to calculate the surface area of each can and determine the cost of materials for each component.
For the can with a radius of 4 inches and a height of 3 inches:
- The area of the top and bottom surfaces is given by A1 = 2πr^2 = 2π(4)^2 = 32π square inches.
- The area of the curved surface is given by A2 = 2πrh = 2π(4)(3) = 24π square inches.
- The total surface area is A = A1 + A2 = 32π + 24π = 56π square inches.
- The cost of the top and bottom material would be 0.80 * A1 = 0.80 * 32π = 25.6π dollars.
- The cost of the curved surface material would be 0.40 * A2 = 0.40 * 24π = 9.6π dollars.
- The total cost of materials for this can would be 25.6π + 9.6π = 35.2π dollars.
For the can with a radius of 3 inches and a height of 4 inches:
- The area of the top and bottom surfaces is given by A1 = 2πr^2 = 2π(3)^2 = 18π square inches.
- The area of the curved surface is given by A2 = 2πrh = 2π(3)(4) = 24π square inches.
- The total surface area is A = A1 + A2 = 18π + 24π = 42π square inches.
- The cost of the top and bottom material would be 0.80 * A1 = 0.80 * 18π = 14.4π dollars.
- The cost of the curved surface material would be 0.40 * A2 = 0.40 * 24π = 9.6π dollars.
- The total cost of materials for this can would be 14.4π + 9.6π = 24π dollars.
Comparing the costs, we see that the can with a radius of 4 inches and a height of 3 inches would cost 35.2π dollars to manufacture while the can with a radius of 3 inches and a height of 4 inches would cost 24π dollars. Therefore, the can with a radius of 4 inches and a height of 3 inches would be more expensive to manufacture.