To solve this problem, we need to find the slant height [tex]\( l \)[/tex] of a regular hexagonal pyramid given the dimensions of the base, its apothem, and the total surface area.
Here is the step-by-step solution:
1. Calculate the Area of the Base:
- The base of the pyramid is a regular hexagon.
- The formula for the area of a regular hexagon is:
[tex]\[
\text{Base Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\][/tex]
- First, find the perimeter of the hexagon:
[tex]\[
\text{Perimeter} = 6 \times \text{side length} = 6 \times 10 = 60 \text{ inches}
\][/tex]
- Substitute the values into the base area formula:
[tex]\[
\text{Base Area} = \frac{1}{2} \times 60 \times 5\sqrt{3} = 30 \times 5\sqrt{3} = 150\sqrt{3} \text{ square inches}
\][/tex]
2. Identify the Total Surface Area Formula:
- The total surface area [tex]\( A \)[/tex] of a regular hexagonal pyramid can be given by:
[tex]\[
A = \text{Base Area} + \frac{1}{2} \times \text{Perimeter} \times \text{Slant Height}
\][/tex]
- We already know:
[tex]\[
A = 420 + 150\sqrt{3} \text{ square inches}
\][/tex]
3. Plug in Known Values and Solve for the Slant Height [tex]\( l \)[/tex]:
- Substitute the values into the surface area formula:
[tex]\[
420 + 150\sqrt{3} = 150\sqrt{3} + \frac{1}{2} \times 60 \times l
\][/tex]
- Simplify and isolate [tex]\( l \)[/tex]:
[tex]\[
420 + 150\sqrt{3} = 150\sqrt{3} + 30l
\][/tex]
[tex]\[
420 = 30l
\][/tex]
[tex]\[
l = \frac{420}{30}
\][/tex]
[tex]\[
l = 14 \text{ inches}
\][/tex]
Based on the calculations, the slant height [tex]\( l \)[/tex] of the hexagonal pyramid is 14 inches. Therefore, the correct answer is:
B. 14 inches
