A regular decagon is a shape with 10 sides of identical length. A piece of chimney pipe has open ends shaped like decagonal bases with sides that are 6.5 inches. The prism is 15 inches tall. What area of aluminum is needed to construct this piece of chimney?
a. 975 in² b. 780 in²
c. 1073 in² d. 878 in²
To find the area of the aluminum needed to construct the piece of chimney, we need to calculate the surface area of all the sides of the decagonal base and the surface area of the rectangular side.
The formula for the area of a regular decagon is given by:
A = (5/4) * s^2 * √(5 + 2√5)
where s is the length of the side of the decagon.
In this case, s = 6.5 inches.
Using the formula, we can find the area of one side of the decagonal base:
A_decagon = (5/4) * (6.5)^2 * √(5 + 2√5)
A_decagon ≈ 87.8 in²
Since there are 10 sides in the decagonal base, the total area of the decagonal base is:
A_total_decagon = 10 * A_decagon
A_total_decagon ≈ 878 in²
Now, let's calculate the area of the rectangular side. The formula for the area of a rectangle is:
A_rectangle = length * width
In this case, the length of the rectangular side is the height of the prism, which is 15 inches, and the width is the perimeter of the decagonal base, which is 10 * 6.5 = 65 inches.
A_rectangle = 15 * 65 = 975 in²
Lastly, to find the total area of aluminum needed to construct the chimney piece, we add the area of the decagonal base and the area of the rectangular side:
Total area = A_total_decagon + A_rectangle
Total area ≈ 878 + 975 = 1853 in²
Therefore, none of the given options (a, b, c, d) are the correct answer.