Answer:
approximately 242.4 cubic inches.
Explanation:
The volume of a prism is given by the formula:
Volume = Base area x Height
We are given that the base area of the prism is 21 square inches, and the height is 5 inches. We need to find the length of one side of the base to determine the base shape of the prism.
The perimeter of the base is 29 inches, so if the base has n sides of equal length, then the length of each side is:
Perimeter = n x Length of one side
29 inches = n x Length of one side
Since we don't know the number of sides, we can't solve for the length of one side directly. However, we can use the fact that the base area is 21 square inches to write an equation involving the length of one side:
Base area = (1/2) x Perimeter x Apothem
21 square inches = (1/2) x 29 inches x Apothem
where the apothem is the distance from the center of the base to the midpoint of a side.
Simplifying this equation, we get:
Apothem = 42/29 inches
Now we can use the apothem to find the length of one side of the base:
Length of one side = 2 x Apothem / √3
Length of one side = 2 x (42/29) inches / √3
Length of one side = 12/√3 inches
Now we can calculate the volume of the prism:
Volume = Base area x Height
Volume = 21 square inches x 5 inches x (12/√3) inches
Volume = 420/√3 cubic inches
Volume ≈ 242.4 cubic inches (rounded to one decimal place)
Therefore, the volume of the prism is approximately 242.4 cubic inches.