Question

brian wants to fence in his triangular plot of farm land that measures 1.1 by 1.5 by 2.2 miles. determine the angles at which the fences of the three sides will meet. rounding each angle to the nearest degree: m∠a ≈ degrees done a triangle with vertices labeled a, b, and c. side ab is 1.5 miles. side ac is 1.1 miles. side bc is 2.2 miles.

Explanation:

Step1: Identify sides for Law of Cosines

We use the Law of Cosines to find angle \( A \). The sides adjacent to angle \( A \) are \( b = 1.1 \) (AC), \( c = 1.5 \) (AB), and the side opposite angle \( A \) is \( a = 2.2 \) (BC). The Law of Cosines formula for angle \( A \) is:
\[
\cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}
\]

Step2: Substitute the values

Substitute \( b = 1.1 \), \( c = 1.5 \), and \( a = 2.2 \) into the formula:
\[
\cos A=\frac{(1.1)^{2}+(1.5)^{2}-(2.2)^{2}}{2\times1.1\times1.5}
\]
First, calculate the numerator:
\[
(1.1)^{2}=1.21, \quad (1.5)^{2}=2.25, \quad (2.2)^{2}=4.84
\]
\[
1.21 + 2.25-4.84=3.46 - 4.84=- 1.38
\]
Then, calculate the denominator:
\[
2\times1.1\times1.5 = 3.3
\]
So,
\[
\cos A=\frac{-1.38}{3.3}\approx - 0.4182
\]

Step3: Find angle \( A \)

To find angle \( A \), take the arccosine of \( - 0.4182 \):
\[
A=\arccos(-0.4182)\approx114.7^{\circ}\approx115^{\circ}
\]

Answer:

\( 115 \)