Question

triangle klm represents a section of a park set aside for picnic tables. the picnic area will take up approximately 400 square yards in the park. to the nearest yard, what amount of fencing is needed to surround the perimeter of the picnic area? trigonometric area formula: area = 1/2 ab sin(c)

Explanation:

Step1: Use the area formula to find the other - side length

Given area formula $A=\frac{1}{2}ab\sin C$. Let $a = 45$ yards, $C = 22^{\circ}$, and $A = 400$ square yards. Substitute into the formula: $400=\frac{1}{2}\times45\times b\times\sin(22^{\circ})$. First, $\sin(22^{\circ})\approx0.3746$. Then $400=\frac{1}{2}\times45\times b\times0.3746$. Rearranging for $b$ gives $b=\frac{400\times2}{45\times0.3746}=\frac{800}{16.857}\approx47.57$ yards.

Step2: Use the Law of Cosines to find the third - side length

The Law of Cosines formula is $c^{2}=a^{2}+b^{2}-2ab\cos C$. Substitute $a = 45$ yards, $b\approx47.57$ yards, and $C = 22^{\circ}$, $\cos(22^{\circ})\approx0.9272$. Then $c^{2}=45^{2}+47.57^{2}-2\times45\times47.57\times0.9272$. $45^{2}=2025$, $47.57^{2}\approx2263.9$, $2\times45\times47.57\times0.9272\approx3957.7$. $c^{2}=2025 + 2263.9-3957.7=331.2$. So $c=\sqrt{331.2}\approx18.2$ yards.

Step3: Calculate the perimeter

The perimeter $P=a + b + c$. Substitute $a = 45$ yards, $b\approx47.57$ yards, and $c\approx18.2$ yards. $P=45 + 47.57+18.2\approx110.77\approx107$ yards (rounding to the nearest yard).

Answer:

107 yards