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 = $\frac{1}{2}absin(c)$

Explanation:

Step1: Find side length \(b\) using area formula

Given area \(A = 65\) square - feet, \(a = 13\) ft, \(C=50^{\circ}\), and area formula \(A=\frac{1}{2}ab\sin(C)\).
Substitute values: \(65=\frac{1}{2}\times13\times b\times\sin(50^{\circ})\).
First, simplify the left - hand side of the equation: \(\frac{1}{2}\times13\times b\times\sin(50^{\circ}) = 6.5b\sin(50^{\circ})\).
Then, solve for \(b\): \(b=\frac{65}{6.5\sin(50^{\circ})}\).
Since \(\sin(50^{\circ})\approx0.766\), \(b=\frac{65}{6.5\times0.766}=\frac{65}{4.979}\approx13\) ft.

Step2: Calculate the perimeter \(P\)

The perimeter of a triangle \(P=a + b+ c\), where \(a = 13\) ft, \(b\approx13\) ft, and \(c = 10\) ft.
\(P=13 + 13+10=36\) ft.
The closest value to \(36\) ft among the options is \(40\) feet.

Answer:

C. 40 feet