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 flower bed?
options: 17 feet, 33 feet, 40 feet, 50 feet
triangle abc has sides: ab = 13 ft, bc = 10 ft, angle at c is 50°. trigonometric area formula: area = \\(\frac{1}{2}ab\sin(c)\\)

Explanation:

Step1: Use the area formula to find side \( b \)

The trigonometric area formula is \( \text{Area} = \frac{1}{2}ab\sin(C) \). Here, \( a = 13 \) ft, \( c = 10 \) ft, \( \angle C = 50^\circ \), and Area = 65.1 sq ft. Wait, actually, let's correct: the two sides with known lengths are 13 ft (AB) and 10 ft (BC), and the included angle? Wait, no, the area formula is \( \text{Area} = \frac{1}{2} \times AB \times BC \times \sin(\angle B) \)? Wait, no, the diagram shows \( AB = 13 \) ft, \( BC = 10 \) ft, angle at C is \( 50^\circ \). Wait, maybe the two sides are 13 and \( b \), with included angle 50? Wait, the area formula given is \( \text{Area} = \frac{1}{2}ab\sin(C) \). Let's assume \( a = 13 \), \( c = 10 \), angle at C is 50? Wait, no, let's use the formula: \( 65.1 = \frac{1}{2} \times 13 \times b \times \sin(50^\circ) \). Wait, maybe the sides are 10 and 13, and the included angle? Wait, let's solve for \( b \). Let's rearrange the formula: \( b = \frac{2 \times \text{Area}}{13 \times \sin(50^\circ)} \). Calculate \( \sin(50^\circ) \approx 0.7660 \). Then \( 2 \times 65.1 = 130.2 \). \( 13 \times 0.7660 \approx 9.958 \). So \( b \approx \frac{130.2}{9.958} \approx 13.07 \), approximately 13? Wait, no, maybe I mixed up the sides. Wait, the triangle has sides AB = 13, BC = 10, and AC = \( b \), angle at C is 50 degrees. Wait, the area formula is \( \frac{1}{2} \times AB \times AC \times \sin(\angle A) \)? No, the formula is \( \frac{1}{2}ab\sin(C) \), where \( a \) and \( b \) are two sides, and \( C \) is the included angle. So if sides are 10 and \( b \), included angle 50, and area 65.1. Then \( 65.1 = \frac{1}{2} \times 10 \times b \times \sin(50^\circ) \). Then \( 65.1 = 5b \times 0.7660 \). \( 5 \times 0.7660 = 3.83 \). So \( b = \frac{65.1}{3.83} \approx 17 \). Ah, that makes sense. So \( b \approx 17 \) ft.

Step2: Calculate the perimeter

Perimeter is the sum of all sides: 13 + 10 + 17 = 40? Wait, 13 + 10 + 17 = 40? Wait, 13 + 10 is 23, plus 17 is 40. Wait, but let's check again. Wait, if \( b \approx 17 \), then sides are 13, 10, 17. Sum is 13 + 10 + 17 = 40. Wait, but the options are 17, 33, 40, 50. So 40 is an option. Wait, but let's verify the area with \( b = 17 \), sides 10 and 17, angle 50. Area = 0.5 10 17 sin(50°) ≈ 0.5 170 0.7660 ≈ 85 0.7660 ≈ 65.11, which matches the given area (65.1). Perfect. So the perimeter is 13 + 10 + 17 = 40.

Answer:

40 feet (the option with 40 feet)