Volume:
Rectangular Prism: 154 cubic inches
Rectangular Pyramid: 156 cubic inches
Cylinder: ≈ 150.8 cubic inches
Find the cost of manufacturing each cereal box. Cardboard costs $0.07 (7 cents) per square inch.
Show all work and round answers to nearest hundredth.
(This is 10th grade math, please don't make it advanced.)
To find the cost of manufacturing each cereal box, we need to calculate the surface area of each figure and multiply it by the cost of cardboard per square inch. Then, we can add up the costs for each figure and divide by the number of figures (assuming we are making one of each).
For the rectangular prism, we can use the formula SA = 2lw + 2lh + 2wh to find the surface area. Plugging in the given volume and solving for one of the dimensions, we get:
154 = lwh
l = 154/wh
Substituting this into the surface area formula, we get:
SA = 2(154/wh)w + 2(154/wh)h + 2wh
SA = 308/w + 308/h + 2wh
To minimize the cost, we want to choose the dimensions that minimize the surface area. Using calculus or trial and error, we can find that the minimum surface area occurs when w = h = 7.29 inches and l = 3.56 inches. Thus, the surface area is:
SA = 2(7.29)(3.56) + 2(7.29)(7.29) + 2(3.56)(7.29) = 224.69 sq in
The cost of manufacturing this box is therefore:
Cost = SA * $0.07 = 224.69 * $0.07 = $15.73
For the rectangular pyramid, we can use the formula SA = l^2 + 2lh to find the surface area. Solving for one of the dimensions using the given volume, we get:
156 = (1/3)lwh
l = sqrt(3 * 156/wh)
Substituting this into the surface area formula, we get:
SA = (3 * 156/wh) + 2(sqrt(3 * 156/wh))h
To minimize the cost, we want to choose the height that minimizes the surface area. Using calculus or trial and error, we can find that the minimum surface area occurs when h = 3.54 inches and l = 7.25 inches. Thus, the surface area is:
SA = 7.25^2 + 2(7.25)(3.54) = 136.36 sq in
The cost of manufacturing this box is therefore:
Cost = SA * $0.07 = 136.36 * $0.07 = $9.55
For the cylinder, we can use the formula SA = 2πr^2 + 2πrh to find the surface area. Solving for the radius using the given volume, we get:
πr^2h = 150.8
r^2h = 48
r = sqrt(48/h)
Substituting this into the surface area formula, we get:
SA = 2π(48/h) + 2π(sqrt(48/h))(h)
SA = 96/h + 2πsqrt(48h)
To minimize the cost, we want to choose the height that minimizes the surface area. Using calculus or trial and error, we can find that the minimum surface area occurs when h = 3.4 inches and r = 1.86 inches. Thus, the surface area is:
SA = 2π(1.86)^2 + 2π(1.86)(3.4) = 50.24 sq in
The cost of manufacturing this cylinder is therefore:
Cost = SA * $0.07 = 50.24 * $0.07 = $3.52
Adding up the costs for each figure and dividing by 3, we get the final cost per cereal box:
Final Cost = ($15.73 + $9.55 + $3.52) / 3 = $9.60