A candy company has hired you as their new production manager. Your job is to choose the form of packaging for a new product from the five choices below. The package must hold at least 100 cubic inches. The cost of the cardboard for the packaging is $0.04 per square inch.
Your project should 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. Make sure to use the formulas given in your lessons and round your answers to the nearest hundredth.
Solid 1:
SHape;
(3Drectangle w;10 b;3 h;6)
Solid 2:
shape;
(pyramid with9 inch base edges and 5 in height)
Solid 3:
Shape;
(cylinder with 2 inch radius and 7 in height)
so, using the formulas, what do you get?
To determine the volume, total surface area, and materials cost for each solid, we can use the formulas for each specific shape.
Solid 1:
Shape: 3D rectangle with width (w) = 10 inches, breadth (b) = 3 inches, and height (h) = 6 inches.
Volume Formula for 3D Rectangle = b * w * h
Using the given values:
Volume = 3 * 10 * 6 = 180 cubic inches.
Surface Area Formula for 3D Rectangle = 2(b*w + b*h + w*h)
Using the given values:
Surface Area = 2(3*10 + 3*6 + 10*6) = 204 square inches.
Materials Cost = Surface Area * Cost per Square Inch
Using the given cost:
Materials Cost = 204 * 0.04 = $8.16.
Solid 2:
Shape: Pyramid with a base edge length of 9 inches and a height of 5 inches.
Volume Formula for Pyramid = (1/3) * base_area * height
Using the given values:
Base Area = 9 * 9 = 81 square inches
Volume = (1/3) * 81 * 5 = 135 cubic inches.
Surface Area Formula for Pyramid = base_area + (1/2) * perimeter_base * slant_height
Using the given values:
Perimeter Base = 4 * base_edge_length = 4 * 9 = 36 inches
Slant Height = square_root((base_edge_length/2)^2 + height^2) = square_root((9/2)^2 + 5^2) = square_root(63.25) ≈ 7.96 inches
Surface Area = 81 + (1/2) * 36 * 7.96 = 231.96 square inches.
Materials Cost = Surface Area * Cost per Square Inch.
Using the given cost:
Materials Cost = 231.96 * 0.04 = $9.28.
Solid 3:
Shape: Cylinder with a radius (r) of 2 inches and a height (h) of 7 inches.
Volume Formula for Cylinder = π * r^2 * h
Using the given values and approximating π to 3.14:
Volume = 3.14 * 2^2 * 7 ≈ 87.92 cubic inches.
Surface Area Formula for Cylinder = 2 * π * r * h + 2 * π * r^2
Using the given values and approximating π to 3.14:
Surface Area = 2 * 3.14 * 2 * 7 + 2 * 3.14 * 2^2 ≈ 87.92 square inches.
Materials Cost = Surface Area * Cost per Square Inch.
Using the given cost:
Materials Cost = 87.92 * 0.04 = $3.52.
Therefore, the volume, total surface area, and materials cost for each solid are as follows:
Solid 1: Volume = 180 cubic inches, Surface Area = 204 square inches, Materials Cost = $8.16.
Solid 2: Volume ≈ 135 cubic inches, Surface Area ≈ 231.96 square inches, Materials Cost = $9.28.
Solid 3: Volume ≈ 87.92 cubic inches, Surface Area ≈ 87.92 square inches, Materials Cost = $3.52.