Use the image to answer the question.
An illustration shows an unfolded version of a pentagonal prism.
A horizontal rectangle is divided into five segments of vertical length 6 feet. The segments are divided with dashed lines. Two pentagons adjoin the top and bottom sides of the second rectangle, also with common dashed lines. Each side of the pentagon is 4 feet. The perpendicular height of the pentagon is drawn with a dotted line extending from the from the center of a side to the center of the pentagon, and is labeled as 5 feet. A right angle symbol is shown where the perpendicular height meets the side.
Find the lateral surface area for the regular pentagonal prism represented by the given net.
(1 point)
To find the lateral surface area of the pentagonal prism, we first need to calculate the area of one of the pentagon faces. The formula for the area of a regular pentagon is given by:
Area of a pentagon = (5/4) * s^2 * cot(π/5)
Where s is the length of the side of the pentagon. In this case, s = 4 feet. Plugging in the values, we get:
Area of a pentagon = (5/4) * 4^2 * cot(π/5)
Area of a pentagon = (5/4) * 16 * cot(π/5)
Area of a pentagon = 20 * cot(π/5)
Now, we need to calculate the lateral surface area of the prism, which would be the sum of the areas of the five rectangular faces. Since the top and bottom faces are pentagons, we only need to consider the three rectangular faces.
The rectangle has a height of 6 feet and a width of 4 feet. Therefore, the area of one rectangular face is:
Area of a rectangular face = 6 * 4 = 24 square feet
Since there are three identical rectangular faces, the total area of these faces is:
Total area of rectangular faces = 3 * 24 = 72 square feet
Finally, we add the area of the two pentagon faces to get the total lateral surface area of the pentagonal prism:
Total lateral surface area = Total area of rectangular faces + 2 * Area of pentagon
Total lateral surface area = 72 + 2 * 20 * cot(π/5) = 72 + 40 * cot(π/5) square feet
Therefore, the lateral surface area of the pentagonal prism represented by the given net is 72 + 40 * cot(π/5) square feet.