Assignment: Optimizing Surface Area and Volume

Dairy King is creating a new package for its1350 mL (1350 cm³) ice cream products. The container for this product will be made of a wax coated cardboard. They need to choose between a cylindrical and rectangular prism shape for the container that will minimize the cost for the product container.

(NOTE: 1 cm³= 1 mL, and 1 m² = 10,000 cm²)

ROUND ALL ANSWERS TO TWO DECIMAL PLACES

Part A:

1. Determine the dimensions of thecylindrical container that minimizes the amount of cardboard (and thus the cost) needed for the container. Then sketch the cylindrical container and label the dimensions on it. Finally, calculate the amount of cardboard needed for the container with these dimensions.

HINT: Which formulas maximize volume, while minimizing surface area, and what relationship in a cylinder will always maximize volume?

Dimension Calculations:	Sketch:
Container has a radius of __________ and a height of _______________.
Surface Area Calculations:
Container has a surface area of __________________ cm2 or _________________ m2

2. If the cardboard costs $0.89 per m², determine the cost to make one container.

Part B:Determine the dimensions of arectangular prism container that would minimize the amount of cardboard (and thus the cost) needed to create it. Sketch the container and label the dimensions on it. Finally, calculate the amount of cardboard needed to create this container.

HINT: Which formulas maximize volume, while minimizing surface area, and what relationship in a cylinder will always maximize volume?

Dimension Calculations:	Sketch:
Container has a side length of ________.
Surface Area Calculations:
Container has a surface area of __________________ cm2 or _________________ m2

2. If the cardboard costs $0.89 per m², determine the cost to make one container.

Conclusion: Describe which container you would recommend to Dairy King for their new container, andwhy. Remember the company will want to minimize its cost. However, if you wish to argue that one container is better for other reasons, feel free to do so.

Part C: Dairy King decides to go with a cylindrical container. However, they wish now to have a plastic top with a waxed coated container. Modify your Surface Area calculations and determine how much it costs to make each container with this new material. (The cardboard still costs $0.89/m², and the plastic costs $1.10/m²).

HINT: What shapes are you dealing with for the container and the lid?

Cylinder Surface Area (Cardboard)	Rectangular Prism Surface Area (Cardboard)
Cost (Cardboard)	Cost (Cardboard)
Cylinder Surface Area (Plastic)	Rectangular Prism Surface Area (Plastic)
Cost (Plastic)	Cost (Plastic)
Cylinder Total Cost	Rectangular Prism Total Cost

Conclusion: Is your recommendation still the same? Why or why not? Explain fully.