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