Question

select the correct answer.
christi is using a display box shaped like a regular pentagonal prism as a gift box. about how much gift wrap does she need to completely cover the box?
image of a blue regular pentagonal prism with height 20 cm and a base edge (or related dimension) labeled 8 cm
a. ( 480 , \text{cm}^2 )
b. ( 800 , \text{cm}^2 )
c. ( 1,020 , \text{cm}^2 )
d. ( 1,600 , \text{cm}^2 )

Explanation:

Step1: Recall the surface area formula for a regular pentagonal prism

The surface area \( S \) of a regular pentagonal prism is given by \( S = 2B + Ph \), where \( B \) is the area of the regular pentagonal base, \( P \) is the perimeter of the base, and \( h \) is the height (or length) of the prism.

For a regular pentagon with side length \( s \), the area of the base \( B \) can be approximated (or calculated using the formula for the area of a regular polygon \( B=\frac{1}{2} \times \text{perimeter of base} \times \text{apothem} \)). But in this case, we can also use a simpler approach. However, maybe we can notice that the problem might be using a simplified version or we can calculate the lateral surface area and the area of the two bases.

First, let's find the perimeter of the regular pentagonal base. A regular pentagon has 5 equal sides. From the diagram, the side length \( s = 8 \) cm. So the perimeter \( P = 5\times s=5\times8 = 40 \) cm.

The height of the prism (the length of the rectangular faces) \( h = 20 \) cm.

The lateral surface area (the area of the 5 rectangular faces) is \( Ph=40\times20 = 800 \) \( \text{cm}^2 \).

Now, we need to find the area of the two pentagonal bases. Let's assume that the apothem (the distance from the center of the pentagon to the midpoint of a side) or we can use the fact that for a regular pentagon, the area of one base \( B \) can be calculated, but maybe in this problem, we can estimate or use the fact that the area of each pentagonal base is non - zero, so the total surface area should be more than the lateral surface area (800 \( \text{cm}^2 \)).

Let's calculate the area of a regular pentagon with side length \( s = 8 \) cm. The formula for the area of a regular pentagon is \( B=\frac{5s^{2}}{4\tan(\frac{\pi}{5})}\). Let's compute \( \tan(\frac{\pi}{5})=\tan(36^{\circ})\approx0.7265 \)

\( B=\frac{5\times8^{2}}{4\times0.7265}=\frac{5\times64}{2.906}=\frac{320}{2.906}\approx110.12 \) \( \text{cm}^2 \)

Then the area of two bases is \( 2B = 2\times110.12 = 220.24 \) \( \text{cm}^2 \)

Now, the total surface area \( S=2B + Ph=220.24+800 = 1020.24\approx1020 \) \( \text{cm}^2 \)

Step2: Analyze the options

• Option A: 480 \( \text{cm}^2 \) is too small, as the lateral surface area alone is 800 \( \text{cm}^2 \)
• Option B: 800 \( \text{cm}^2 \) is just the lateral surface area, we need to add the area of the two bases, so it's too small.
• Option C: 1020 \( \text{cm}^2 \) matches our calculation of the total surface area (lateral surface area + area of two bases)
• Option D: 1600 \( \text{cm}^2 \) is too large.

Answer:

C. \( 1,020\ \text{cm}^2 \)