Answer:
\(SA\approx 446.55\text{ ft}^2\)
Step-by-step explanation:
We can see that this pyramid is composed of:
- hexagonal base
- 6 isosceles triangle sides
So, an equation that models its surface area is:
\(SA = A_{\rm base} + 6A_{\rm triangle}\)
Plugging in the given base area and the area of a triangle formula:
- \(A_{\rm base} = 50\text{ ft}^2\)
- \(A_{\rm triangle} = \dfrac{1}{2} bh\)
↓↓↓
\(SA = 50\text{ ft}^2 + 6(1/2)(8\text{ ft})(h)\)
\(SA = 50\text{ ft}^2 + (24\text{ ft})h\)
We can solve for the triangles' height using the Pythagorean Theorem:
\(a^2 + b^2 = c^2\)
↓ plugging in the given values
\((8/2)^2 + h^2 = 17^2\)
↓ solving for h
\(h^2 = 273\)
↓ taking the square root of both sides
\(h \approx 16.523\text{ ft}\)
Finally, we can plug this value into the surface area equation:
\(SA \approx 50\text{ ft}^2 + (24\text{ ft})(16.523\text{ ft})\)
\(\boxed{SA\approx 446.55\text{ ft}^2}\)
