Let's solve the problem step by step to find the slant height of the regular hexagonal pyramid.
1. Calculate the area of the base:
The base of the pyramid is a regular hexagon with a side length of 10 inches. The area of a regular hexagon can be calculated using the formula:
[tex]\[
\text{Area of base} = \frac{3 \sqrt{3}}{2} s^2
\][/tex]
where [tex]\( s \)[/tex] is the side length. Substituting [tex]\( s = 10 \)[/tex]:
[tex]\[
\text{Area of base} = \frac{3 \sqrt{3}}{2} \cdot 10^2 = \frac{3 \sqrt{3}}{2} \cdot 100 = 150 \sqrt{3} \text{ square inches}
\][/tex]
2. Calculate the total surface area:
Given the total surface area [tex]\( A \)[/tex] is [tex]\( 420 + 150 \sqrt{3} \)[/tex] square inches. We know the total surface area [tex]\( A \)[/tex] is the sum of the lateral surface area [tex]\( A_L \)[/tex] and the area of the base [tex]\( A_B \)[/tex]:
[tex]\[
A = A_L + A_B
\][/tex]
So,
[tex]\[
A_L = A - A_B = (420 + 150 \sqrt{3}) - (150 \sqrt{3}) = 420 \text{ square inches}
\][/tex]
3. Calculate the perimeter of the base:
The perimeter [tex]\( P \)[/tex] of the hexagonal base can be calculated by:
[tex]\[
P = 6s
\][/tex]
where [tex]\( s = 10 \)[/tex]. So,
[tex]\[
P = 6 \cdot 10 = 60 \text{ inches}
\][/tex]
4. Find the slant height:
The lateral surface area [tex]\( A_L \)[/tex] of the pyramid is given by the formula:
[tex]\[
A_L = \frac{1}{2} \times P \times \text{slant height}
\][/tex]
We already have [tex]\( A_L = 420 \)[/tex] square inches and [tex]\( P = 60 \)[/tex] inches. Rearranging the formula to solve for the slant height:
[tex]\[
\text{slant height} = \frac{2 A_L}{P} = \frac{2 \times 420}{60}
\][/tex]
[tex]\[
\text{slant height} = \frac{840}{60} = 14 \text{ inches}
\][/tex]
Therefore, the slant height of the regular hexagonal pyramid is [tex]\( \boxed{14} \)[/tex] inches. The correct answer is B. 14 inches.
