Question

You are hired to help minimize production costs. Your job is to choose the form of packaging for a new product from the five choices below. The package must hold at least 900 cubic inches. The cost of the cardboard for the packaging is $0.03 per square inch. Include the volume, total surface area, and materials cost for each solid given below, including the formulas you used and each step of your work.

Answer 1

Solid 1: Square Prism with each side of the base equal to 10 in. and a height of 9 in.

Solid 2: Square Pyramid with each side of the base equal to 12 in. and a height of 19 in.

Solid 3: Cylinder with a radius of 6 in. and a height of 8 in.

Solid 4: Cone with a radius of 9 in. and a height of 11 in.

Solid 5: Sphere with a radius of 6 in.

Answer 2

Logically, the answer is the sphere, as it is the figure which gives maximum volume for the same total surface area. But I'll just solve them like you want it.
I'm just writing the numerical values without the units.Please resolve

Solid 1: Square Prism with each side of the base equal to 8 in. and a height of 8 in.
Volume = 8^3 = 512
Area = 8^2 * 6 = 384
V/A Ratio = 1.33 (We need the highest ratio, that's why they hired us)
Cost = $7.68

Solid 2: Square Pyramid with each side of the base equal to 10 in. and a height of 15 in.
Volume = 1/3 * 10^2 * 15 = 500
Slant height = [(10/2)^2 + 15^2]^(1/2) = root of 250 = 15.81
Area = 2*10*15.81 + 10^2 = 416.23
V/A Ratio = 1.20
Cost = $8.34

Solid 3: Cylinder with a radius of 4 in. and a height of 10 in.
Volume = pi*4*4*10 = 502.65
Area = 2*pi*4*4 + 2*pi*4*10 = 351.86
V/A Ratio = 1.43
Cost = $7.04

Solid 4: Cone with a radius of 7 in. and a height of 10 in.
Volume = (1/3)*pi*7*7*10 = 513.13
Slant height = [(7^2)+(10^2)]^(1/2) = 12.21
Area = pi*7*12.21 + pi*7*7 = 422.37
V/A Ratio = 1.21
Cost = $8.45

Solid 5: Sphere with a radius of 5 in.
Volume = (4/3)*pi*(5^3) = 523.60
Area = 4*pi*(r^2) = 314.16
V/A Ratio = 1.67
Cost = $6.28

Hence, Solid 5 must be the packaging model opted for

Hope this helps :)