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?
You've posted this before.
Have you found the surface area of the cylinder?
Have you found the surface area of the rectangular prism?
I'll be glad to check those answers.
To find the cost of producing each type of container, we need to calculate the surface area of each container and then multiply by the cost per square foot of aluminum.
For the cylindrical container:
1. The base of the cylinder is a circle, so the area of the base can be calculated using the formula for the area of a circle: A = π * r^2, where r is the radius.
2. The radius of the cylinder is half of the diameter, so the radius is 1 ft / 2 = 0.5 ft.
3. Using the formula, the area of the base is A = π * (0.5 ft)^2 = π * 0.25 ft^2.
4. The curved surface area of the cylinder is the circumference of the base (2πr) multiplied by the height of the cylinder. The height is given as 1.25 ft.
5. So, the curved surface area of the cylinder is A = 2π * 0.5 ft * 1.25 ft = 2.5π ft^2.
6. Finally, to get the total surface area, we add the base area and the curved surface area: Total surface area = base area + curved surface area = π * 0.25 ft^2 + 2.5π ft^2 = 2.75π ft^2.
For the rectangular prism:
1. The surface area of a rectangular prism can be calculated using the formula: A = 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height.
2. Plugging in the given values, the surface area is A = 2 * 0.75 ft * 0.75 ft + 2 * 0.75 ft * 1 ft + 2 * 0.75 ft * 1 ft = 2.25 ft^2 + 1.5 ft^2 + 1.5 ft^2 = 4.25 ft^2.
Now, we can calculate the cost of producing each container by multiplying the surface area by the cost per square foot of aluminum.
Cost of producing the cylindrical container:
1. The surface area of the cylindrical container is 2.75π ft^2.
2. The cost per square foot of aluminum is given as $0.02.
3. So, the total cost to produce the cylindrical container is 2.75π ft^2 * $0.02/ft^2 ≈ $0.055π ≈ $0.17 (rounded to the nearest cent).
Cost of producing the rectangular prism container:
1. The surface area of the rectangular prism container is 4.25 ft^2.
2. The cost per square foot of aluminum is given as $0.02.
3. So, the total cost to produce the rectangular prism container is 4.25 ft^2 * $0.02/ft^2 = $0.085.
To determine which container holds more, we can compare their volumes because the question does not specify any other criteria.
The volume of the cylindrical container can be calculated using the formula for the volume of a cylinder: V = π * r^2 * h.
The volume of the cylindrical container is V = π * (0.5 ft)^2 * 1.25 ft = 0.625π ft^3.
The volume of the rectangular prism container is given by multiplying its length, width, and height: V = 0.75 ft * 0.75 ft * 1 ft = 0.5625 ft^3.
Comparing the volumes, we find that the cylindrical container holds more aluminum (0.625π ft^3) compared to the rectangular prism container (0.5625 ft^3).