A company is making two types of aluminum containers. One is a cylinder with a height of 1.25 feet and a diameter of 1 foot. The other is a rectangular prism with a length of 0.75 foot, a width of 0.75 foot, and a height of 1 foot. Aluminum costs $.02 per square foot. How much will it cost to produce each type of container? Round to the nearest cent. Which container holds more?
The Cylinder
As = 2*pi*r^2 + pi*D*h. = Surface area.
V = pi*r^2 * h.
Cost = As * 0.02/Ft^2.
The Rectangular Prism.
As = 2(L*h + W*h + L*W). = Surface area
V = L * W * h.
Cost = As * 0.02/Ft^2.
NOTES
D = Diameter.
The container with the largest volume
holds the most.
To find the cost of producing each type of container, we need to calculate the surface area of each shape and multiply it by the cost of aluminum per square foot.
1. Cylindrical Container:
The surface area of a cylinder consists of two circles (top and bottom) and the curved surface. The formula for the curved surface area of a cylinder is given by A = 2πrh, where π is a constant (approximately 3.14), r is the radius, and h is the height.
Given information:
- Diameter = 1 foot
- Radius (r) = diameter/2 = 1/2 = 0.5 foot
- Height (h) = 1.25 foot
Calculating the curved surface area:
A = 2 * 3.14 * 0.5 * 1.25
A ≈ 3.925 square feet
The cost to produce the cylindrical container:
Cost = A * Cost per square foot = 3.925 * $0.02
Cost ≈ $0.0785 ≈ $0.08 (rounded to the nearest cent)
2. Rectangular Prism Container:
The surface area of a rectangular prism consists of six sides: two pairs of congruent rectangles and two pairs of congruent squares. To find the surface area, we add the areas of these six sides.
Given information:
- Length = 0.75 foot
- Width = 0.75 foot
- Height = 1 foot
Calculating the surface area:
A = 2(length * width) + 2(length * height) + 2(width * height)
A = 2(0.75 * 0.75) + 2(0.75 * 1) + 2(0.75 * 1)
A = 1.125 + 1.5 + 1.5
A = 4.125 square feet
The cost to produce the rectangular prism container:
Cost = A * Cost per square foot = 4.125 * $0.02
Cost = $0.0825 ≈ $0.08 (rounded to the nearest cent)
Therefore, the cost to produce each type of container is approximately $0.08.
To determine which container holds more, we can compare their volumes.
1. Cylindrical Container:
The volume of a cylinder is calculated by multiplying the area of the base (circle) by the height. The formula for the volume of a cylinder is given by V = πr²h, where π is a constant (approximately 3.14), r is the radius, and h is the height.
Given information:
- Diameter = 1 foot
- Radius (r) = diameter/2 = 1/2 = 0.5 foot
- Height (h) = 1.25 foot
Calculating the volume:
V = 3.14 * 0.5² * 1.25
V ≈ 0.98174 cubic feet
2. Rectangular Prism Container:
The volume of a rectangular prism is calculated by multiplying the length, width, and height together. The formula for the volume of a rectangular prism is given by V = lwh, where l is the length, w is the width, and h is the height.
Given information:
- Length = 0.75 foot
- Width = 0.75 foot
- Height = 1 foot
Calculating the volume:
V = 0.75 * 0.75 * 1
V = 0.5625 cubic feet
Therefore, the cylindrical container holds more with a volume of approximately 0.98174 cubic feet (rounded).