Find the cost of manufacturing each cereal box. Cardboard costs $0.07 (7 cents) per square inch. Show all work in the box provided and round answers to nearest hundredth.
Rectangular Prism:
V = 7 in x 11 in x 2 in
V = 154 cubic inches
Rectangular Pyramid:
6 × 6 = 36 sq in
Volume = (1/3) × 36 sq in × 13 in = 156 cubic inches
Cylinder:
πr^2h
= π(2)^2(12)
= 4π(12)
= 48π
≈ 150.8 cubic inches
(This is 10th grade math, please don't make advanced)
Show all work.
Rectangular Prism:
Surface area = 2lw + 2lh + 2wh = 2(7)(11) + 2(7)(2) + 2(11)(2) = 154 + 28 + 44 = 226 sq in
Cost of cardboard = 226 sq in × $0.07/sq in = $15.82
Cost per cereal box = $15.82 ÷ 12 boxes = $1.32/box
Rectangular Pyramid:
Surface area = lw + 2[(1/2)l × s] = 6(6) + 2[(1/2)(6)(5)] = 36 + 30 = 66 sq in
Cost of cardboard = 66 sq in × $0.07/sq in = $4.62
Cost per cereal box = $4.62 ÷ 12 boxes = $0.39/box
Cylinder:
Surface area = 2πr^2 + 2πrh = 2π(2)^2 + 2π(2)(12) = 8π + 48π = 56π sq in
Cost of cardboard = 56π sq in × $0.07/sq in ≈ $12.40
Cost per cereal box = $12.40 ÷ 12 boxes = $1.03/box
Therefore, the cost of manufacturing each cereal box varies depending on the shape of the box and ranges from $0.39 to $1.32 per box.
To find the cost of manufacturing each cereal box, we need to calculate the surface area of each shape and then multiply it by the cost of cardboard.
For the Rectangular Prism:
The volume of the rectangular prism is given as V = 7 in x 11 in x 2 in = 154 cubic inches.
To find the total surface area of the rectangular prism, we need to calculate the area of each face and then add them up.
The rectangular prism has 6 faces, and each face has the same dimensions of 7 in x 11 in.
The total surface area of the rectangular prism is therefore 2(7 in x 11 in) + 2(7 in x 2 in) + 2(11 in x 2 in) = 154 sq in + 28 sq in + 44 sq in = 226 sq in.
Using the given cost of cardboard as $0.07 per square inch, we can calculate the cost of cardboard for the rectangular prism as:
Cost = Surface Area x Cost per square inch
Cost = 226 sq in x $0.07/sq in
Cost ≈ $15.82
Therefore, the cost of manufacturing each cereal box in the shape of a rectangular prism is approximately $15.82.
For the Rectangular Pyramid:
The volume of the rectangular pyramid is given as 156 cubic inches.
To find the surface area of the rectangular pyramid, we only need to calculate the area of the base, as the pyramid has triangular faces that are not made from cardboard.
The base of the rectangular pyramid has dimensions of 6 in x 6 in, so its area is 6 in x 6 in = 36 sq in.
Using the given cost of cardboard as $0.07 per square inch, we can calculate the cost of cardboard for the rectangular pyramid as:
Cost = Base Area x Cost per square inch
Cost = 36 sq in x $0.07/sq in
Cost ≈ $2.52
Therefore, the cost of manufacturing each cereal box in the shape of a rectangular pyramid is approximately $2.52.
For the Cylinder:
The volume of the cylinder is given as approximately 150.8 cubic inches.
To find the surface area of the cylinder, we need to calculate the area of the two bases and the lateral surface area.
The base of the cylinder is a circle with a radius of 2 inches, so its area is given by A = πr^2 = π(2 in)^2 = 4π in^2.
The lateral surface area of the cylinder is given by A = 2πrh, where r is the radius and h is the height of the cylinder. The given height is not provided, so we cannot calculate the lateral surface area.
Since the cost of cardboard is only given per square inch, we cannot calculate the cost of manufacturing each cereal box in the shape of a cylinder without knowing the height of the cylinder.
Therefore, we cannot determine the cost of manufacturing each cereal box in the shape of a cylinder without more information.
In summary:
- The cost of manufacturing each cereal box in the shape of a rectangular prism is approximately $15.82.
- The cost of manufacturing each cereal box in the shape of a rectangular pyramid is approximately $2.52.
- The cost of manufacturing each cereal box in the shape of a cylinder cannot be determined without knowing the height of the cylinder.