Question

select the correct answer. a new toy comes in the shape of a regular hexagonal pyramid. the base has side lengths of 10 inches and the apothem is (5sqrt{3}) inches. if the surface area is (420 + 150sqrt{3}) square inches, what is the slant height?

a. 28 inches

b. 7 inches

c. 11 inches

d. 14 inches

Explanation:

Step1: Recall surface area formula for regular pyramid

The surface area \( S \) of a regular pyramid is the sum of the base area \( B \) and the lateral surface area \( L \). For a regular hexagonal pyramid, the base is a regular hexagon. The formula for the area of a regular hexagon is \( B=\frac{1}{2}\times perimeter\times apothem \). The lateral surface area \( L=\frac{1}{2}\times perimeter\times slant\ height \) (let slant height be \( l \)).

First, calculate the perimeter of the base. A regular hexagon has 6 sides, each of length 10 inches, so perimeter \( P = 6\times10=60 \) inches.

Step2: Calculate the base area \( B \)

Using the formula for the area of a regular hexagon: \( B=\frac{1}{2}\times P\times apothem \). Substitute \( P = 60 \) and apothem \( = 5\sqrt{3} \):
\( B=\frac{1}{2}\times60\times5\sqrt{3}=150\sqrt{3} \) square inches.

Step3: Relate surface area to base and lateral area

We know the total surface area \( S=B + L \), and \( S = 420+150\sqrt{3} \). We already found \( B = 150\sqrt{3} \), so substitute into the surface area formula:
\( 420 + 150\sqrt{3}=150\sqrt{3}+L \)
Subtract \( 150\sqrt{3} \) from both sides: \( L = 420 \) square inches.

Step4: Solve for slant height \( l \)

The formula for lateral surface area of a regular pyramid is \( L=\frac{1}{2}\times P\times l \). We know \( L = 420 \) and \( P = 60 \), so substitute:
\( 420=\frac{1}{2}\times60\times l \)
Simplify the right side: \( 420 = 30l \)
Divide both sides by 30: \( l=\frac{420}{30}=14 \) inches.

Answer:

D. 14 inches