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Bob is standing 25 feet from a lamppost that is to his left and 30 feet from a lamppost that is to his right. The distance between the two lampposts is 20 feet. What is the measure of the angle formed from the line from each lamppost to Bob? Approximate to the nearest degree.

[tex]\[
\begin{array}{l}
1. \quad 20^2 = 25^2 + 30^2 - 2(25)(30) \cos(A) \\
2. \quad 400 = 625 + 900 - (1500) \cos(A) \\
3. \quad 400 = 1525 - (1500) \cos(A) \\
4. \quad -1125 = - (1500) \cos(A) \\
\end{array}
\][/tex]

[tex]\[
A \approx \text{ degrees}
\][/tex]


Sagot :

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].
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