He knows that his left and right boundary lines connect near his house at
an angle of 143°. The left boundary length is 95 m and the right is 110
m. Help the farmer determine the length of his rear property line.
This is an instance of solving a triangle using the law of cosines, which states that for any triangle with sides of length a, b, and c (where c is the side opposite the given angle), the following equation holds: c² = a² + b² - 2ab cos(C).
In this case:
a = 95 m
b = 110 m
C = 143°
We are asked to solve for c, or the rear property line.
c² = (95 m)² + (110 m)² - 2(95 m)(110 m) cos(143°)
cos(143°) is approximately -0.7431448254773942,
so
c² = 9025 m² + 12100 m² - 2(95 m)(110 m)(-0.7431448254773942)
c² = 9025 m² + 12100 m² +15538.70 m²
c² = 36663.70 m²
Taking the square root of both sides to solve for c gives approximately
c ≈ 191.47 m,
So, the length of the rear property line is approximately 191.47 meters.
To determine the length of the rear property line, we can use the law of cosines.
The law of cosines states that for any triangle with sides a, b, and c and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, the left boundary line is 95 m long (a), the right boundary line is 110 m long (b), and the angle between them is 143° (C).
We want to find the length of the rear property line (c).
Let's plug in the given values into the equation:
c^2 = 95^2 + 110^2 - 2 * 95 * 110 * cos(143°)
Let's calculate:
c^2 = 9025 + 12100 - 20900 * cos(143°)
Now, calculate the cosine of 143°:
cos(143°) ≈ -0.766
c^2 = 9025 + 12100 - 20900 * (-0.766)
c^2 = 9025 + 12100 + 20900 * 0.766
c^2 = 9025 + 12100 + 16063.4
c^2 ≈ 37188.4
Now, take the square root of both sides to find the length of the rear property line:
c ≈ √37188.4
c ≈ 192.72
Therefore, the length of the rear property line is approximately 192.72 m.