A company packages it's cereal in a rectangular prism that is 2.5 inches by 7 inches by 12 inches. It is considering packaging it in a cylinder- shaped container having a 6 inch diameter and a height of 7.5 inches. Which used the least amount of packaging? Prove your answer.
To determine which packaging uses the least amount of material, we need to compare the surface areas of the rectangular prism and the cylinder-shaped container.
First, let's calculate the surface area of the rectangular prism:
Given dimensions: length = 2.5 inches, width = 7 inches, height = 12 inches
The surface area of a rectangular prism is given by the formula: A = 2lw + 2lh + 2wh
Plugging in the values, we have:
A = 2(2.5)(7) + 2(2.5)(12) + 2(7)(12)
A = 35 + 60 + 168
A = 263 square inches
Now, let's calculate the surface area of the cylinder-shaped container:
Given dimensions: diameter = 6 inches, height = 7.5 inches
To find the surface area of a cylinder, we need to calculate the lateral surface area and add it to the surface area of the two circular bases.
The lateral surface area of a cylinder is given by the formula: A = 2πrh
Plugging in the values, we have:
A = 2π(3)(7.5)
A ≈ 94.25 square inches (rounded to two decimal places)
The surface area of each circular base is given by the formula: A = πr^2
Plugging in the values, we have:
A = π(3^2)
A = π(9)
A ≈ 28.27 square inches (rounded to two decimal places)
Since there are two circular bases, we multiply the area of one base by 2 to get the total area for both bases:
2(28.27) ≈ 56.54 square inches
Finally, we add the lateral surface area to the sum of the areas of the two bases:
94.25 + 56.54 ≈ 150.79 square inches
Comparing the surface areas, we find that the rectangular prism has a surface area of 263 square inches, while the cylinder-shaped container has a surface area of approximately 150.79 square inches.
Therefore, the cylinder-shaped container uses the least amount of packaging material.
surface area of box
= 2(2.5)(7) + 2(2.5)(12) + 2(7)(12)
= 263 inches^2
surface area of cylinder
= 2πr^2 +2πrh
= 2π(3^2) + 2π(3)(7.5)
= 18π + 45π = 63π
or appr 197.9 inches^2
so what do you think?