Trig tent question

Hello, goldbug78!

Quote:

A two-person tent is to be made so that the height at the center is a = 4 feet.
If the sides of the tent are to meet the ground at an angle 60°, and the tent is to be b = 5 feet in length,
how many square feet of material will be needed to make the tent?
(Assume that the tent has a floor and is closed at both ends, and give your answer in exact form.)

Code:

* * \ * \ * \ /|\ \ / | \ \ x / |4 \ x * / | \ * / | \ * 5 *-----*-----* : x/2 : x/2 :

The front is an equilateral triangle with altitude 4 feet.

Pythagorus: . $\left(\tfrac{x}{2}\right)^2 + 4^2 \:=\:x^2 \quad\Rightarrow\quad \tfrac{x^2}{4} + 16 \:=\:x^2$

. . . . . . . . . . $\tfrac{3}{4}x^2 \:=\:16 \quad\Rightarrow\quad x^2 \:=\:\tfrac{64}{3} \quad\Rightarrow\quad x \:=\:\tfrac{8\sqrt{3}}{3}$

The front triangle has area: . $\tfrac{1}{2}\left(\tfrac{8\sqrt{3}}{3}\right)(4) \:=\:\tfrac{16\sqrt{3}}{3}$
. . The two triangular panels have area: . $2 \times \tfrac{16\sqrt{3}}{3} \:=\:\tfrac{32\sqrt{3}}{3}$

A rectangular panel has area: . $\left(\tfrac{8\sqrt{3}}{3}\right)(5) \:=\:\tfrac{40\sqrt{3}}{3}$
. . The three rectangular panels have area: . $3 \times\tfrac{40\sqrt{3}}{3} \:=\: 40\sqrt{3}$

The total area is: . $\frac{32\sqrt{3}}{3} + 40\sqrt{3} \;=\;\frac{152\sqrt{3}}{3}\text{ ft}^2$