Question

triangle abc is a sketch of a triangular flower bed that has an area of 65.1 square feet. to the nearest foot, what amount of fencing is needed to surround the perimeter of the flower bed? trigonometric area formula: area = 1/2 ab sin(c)

Explanation:

Step1: Use area formula to find side length

Given area formula $A=\frac{1}{2}ab\sin C$. Let $a = 10$, $C = 50^{\circ}$ and $A=65.1$. Substitute into the formula: $65.1=\frac{1}{2}\times10\times b\times\sin(50^{\circ})$. First, simplify the right - hand side: $\frac{1}{2}\times10\times\sin(50^{\circ})\times b = 5\times\sin(50^{\circ})\times b$. Then solve for $b$: $b=\frac{65.1}{5\times\sin(50^{\circ})}$. Since $\sin(50^{\circ})\approx0.766$, $b=\frac{65.1}{5\times0.766}=\frac{65.1}{3.83}\approx17$.

Step2: Use Law of Cosines to find the third side

The Law of Cosines is $c^{2}=a^{2}+b^{2}-2ab\cos C$. Here, $a = 10$, $b\approx17$, and $C = 50^{\circ}$, $\cos(50^{\circ})\approx0.643$. Then $c^{2}=10^{2}+17^{2}-2\times10\times17\times0.643=100 + 289-2\times10\times17\times0.643=100 + 289 - 218.62=170.38$. So $c=\sqrt{170.38}\approx13$.

Step3: Calculate the perimeter

The perimeter $P=a + b + c$. Substitute $a = 10$, $b\approx17$, $c\approx13$. $P=10 + 17+13=40$.

Answer:

40 feet