To determine the measure of the angle formed by the lines from each lamppost to Bob, we will use the Law of Cosines. The given distances are:
- The distance from Bob to the left lamppost: [tex]\( b = 25 \)[/tex] feet
- The distance from Bob to the right lamppost: [tex]\( c = 30 \)[/tex] feet
- The distance between the two lampposts: [tex]\( a = 20 \)[/tex] feet
According to the Law of Cosines:
[tex]\[
a^2 = b^2 + c^2 - 2bc \cos(A)
\][/tex]
1. Substitute the given values into the equation:
[tex]\[
20^2 = 25^2 + 30^2 - 2(25)(30) \cos(A)
\][/tex]
2. Calculate the squares and multiply:
[tex]\[
400 = 625 + 900 - 2(25)(30) \cos(A)
\][/tex]
3. Simplify the equation:
[tex]\[
400 = 1525 - 1500 \cos(A)
\][/tex]
4. Isolate the cosine term:
[tex]\[
400 - 1525 = -1500 \cos(A)
\][/tex]
[tex]\[
-1125 = -1500 \cos(A)
\][/tex]
5. Solve for [tex]\(\cos(A)\)[/tex]:
[tex]\[
\cos(A) = \frac{-1125}{-1500} = 0.75
\][/tex]
6. Calculate the angle [tex]\(A\)[/tex] using the inverse cosine function:
[tex]\[
A = \cos^{-1}(0.75)
\][/tex]
7. The angle [tex]\(A\)[/tex] in radians:
[tex]\[
A \approx 0.7227342478134157 \text{ radians}
\][/tex]
8. Convert the angle from radians to degrees using the conversion [tex]\(180^\circ / \pi\)[/tex]:
[tex]\[
A \approx 41.40962210927086 \text{ degrees}
\][/tex]
9. Round the angle to the nearest degree:
[tex]\[
A \approx 41^\circ
\][/tex]
Thus, the measure of the angle formed by the lines from each lamppost to Bob is approximately [tex]\(41^\circ\)[/tex].
