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Application Project Math 1401-003

Due: April 26, 2021

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You are making a new drink and need to make a can that holds 355 cm3 (about 12 oz.). You are presented with three shapes of can to choose from:

1. A sphere.

2. A cylinder.

3. a rectangular prism with a square base.

There are two factors that you need to consider when choosing which shape to use: the cost to make the can; and the cost to ship the cans. The cost of the can is proportional to the surface area of the can: the larger the surface area the more the can costs to make. The cost to ship the can is proportional to the volume of the box needed to ship 12 cans: the bigger the box the larger the shipping costs.

Use the table below and answer the following questions to determine the can with least surface area:

The Volume The surface area

Sphere 4 3 πr3 4πr2

Cylinder πr2h 2πr2 + 2πrh

Rectangular prism `wh 2`h + 2`w + 2hw

1. (2 points) Use the volume formula of a sphere to solve for the radius. What is the radius of a sphere with a volume of 355 cm2?

2. (2 points) Use the radius from part 1) to find the surface area of the sphere.

3. (2 points) Use the volume formula of a cylinder to solve for the height of a cylinder with a volume of 355 cm2 in terms of its radius.

4. (2 points) Use your equation from part 3) to find a function from the radius of the cylinder to the surface area.

5. (2 points) Sketch a graph of your function from part 4), and find the minimum surface area using calculus. What is the height, radius and surface area of this cylinder?

6. (2 points) Use the volume formula of a rectangular prism to solve for the height of a rectangular prism with a volume of 355 cm2 and a square base in terms of the length of a side of the base.

7. (2 points) Use your equation from part 6) to find a function from the length of a side of the base of the rectangular prism to the surface area.

8. (2 points) Sketch a graph of your function from part 7), and find the minimum surface area using calculus. What is the height, and length of a side of the base and surface area of this rectangular prism?

9. (1 point) Which shape minimizes the surface area?

To pack the shapes, we will treat them like the rectangular prism that they fit insider:

The Volume of the rectangular prism

Sphere (2r)3

Cylinder (2r)2h

Rectangular prism `wh

1. (2 points) What is the volume of the box needed to hold 12 of the spheres from problem 2) above?

2. (2 points) What is the volume of the box needed to hold 12 of the cylinders from problem 5) above?

3. (2 points) What is the volume of the box needed to hold 12 of the rectangular prism from problem 7) above?

4. (1 point) Which shape needs the smallest box?

(6 points) Which shape do you think would be the best for your drink, taking into account both the surface area and volume needed to ship? Why did you choose this shape? Does your conclusion match how other drinks are canned? If your shape does not match how other drinks are canned brefily disuss why you think you have a diffrent shape.

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