Question

the flag is an isosceles triangle. the bottom side of the flag measures 5 feet because the triangle is. because, the trigonometric triangle formula is used to calculate that the area is $\frac{1}{2}(5)(5)sin(36^{circ}) = 7.35$ square feet. recalling that this triangle is isosceles, we can write the area equation to solve for $x$. $=\frac{1}{2}(x)(5)sin()$. solving for $x$ and adding this value to the length of the other sides, the perimeter $approx$ (rounded to the nearest tenth)

Explanation:

Step1: Recall area formula

The area formula for a triangle is $A=\frac{1}{2}ab\sin C$. We know the area $A = 7.35$ square - feet, $a = x$, $b = 5$ and we need to find the included - angle. Since the triangle is isosceles and the non - equal angle is $36^{\circ}$, the other two equal angles are $\frac{180 - 36}{2}=72^{\circ}$. The angle between sides $x$ and $5$ is $72^{\circ}$. So the area equation is $7.35=\frac{1}{2}(x)(5)\sin(72^{\circ})$.

Step2: Solve for $x$

First, we know that $\sin(72^{\circ})\approx0.9511$. The equation $7.35=\frac{1}{2}(x)(5)\sin(72^{\circ})$ can be rewritten as $7.35=\frac{1}{2}(x)(5)(0.9511)$. Then $7.35 = 2.37775x$. Solving for $x$, we get $x=\frac{7.35}{2.37775}\approx3.1$.

Step3: Calculate the perimeter

The sides of the isosceles triangle are $x = 3.1$, $3.1$ and $5$. The perimeter $P=x + x+5=3.1+3.1 + 5=11.2$ feet.

Answer:

$11.2$