Answer :
We need to find the measure of the angle \(A\) formed between the lines from each lamppost to Bob using the Law of Cosines.
Given distances:
- \(a = 25\) feet (distance from Bob to the left lamppost)
- \(b = 30\) feet (distance from Bob to the right lamppost)
- \(c = 20\) feet (distance between the two lampposts)
According to the Law of Cosines:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(A) \][/tex]
Substituting the known values into the equation:
[tex]\[ 20^2 = 25^2 + 30^2 - 2 \cdot 25 \cdot 30 \cdot \cos(A) \][/tex]
Step-by-step solution:
1. Calculate the squares of the lengths:
[tex]\[ 20^2 = 400 \][/tex]
[tex]\[ 25^2 = 625 \][/tex]
[tex]\[ 30^2 = 900 \][/tex]
2. Substitute these values into the equation:
[tex]\[ 400 = 625 + 900 - 2 \cdot 25 \cdot 30 \cdot \cos(A) \][/tex]
3. Combine the constant terms on the right side:
[tex]\[ 400 = 1525 - 1500 \cos(A) \][/tex]
4. Rearrange to solve for \(\cos(A)\):
[tex]\[ 400 = 1525 - 1500 \cos(A) \][/tex]
[tex]\[ 400 - 1525 = -1500 \cos(A) \][/tex]
[tex]\[ -1125 = -1500 \cos(A) \][/tex]
5. Divide both sides by -1500:
[tex]\[ \cos(A) = \frac{-1125}{-1500} \][/tex]
[tex]\[ \cos(A) = 0.75 \][/tex]
Now, we need to find the angle \(A\) by taking the arccosine (inverse cosine) of 0.75.
[tex]\[ A = \arccos(0.75) \][/tex]
This angle in radians:
[tex]\[ A \approx 0.7227 \text{ radians} \][/tex]
Convert this angle to degrees:
[tex]\[ A \approx 41.41 \text{ degrees} \][/tex]
Approximately to the nearest degree:
[tex]\[ A \approx 41^\circ \][/tex]
Thus, the measure of the angle formed from the line from each lamppost to Bob is [tex]\( \boxed{41} \)[/tex] degrees.
Given distances:
- \(a = 25\) feet (distance from Bob to the left lamppost)
- \(b = 30\) feet (distance from Bob to the right lamppost)
- \(c = 20\) feet (distance between the two lampposts)
According to the Law of Cosines:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(A) \][/tex]
Substituting the known values into the equation:
[tex]\[ 20^2 = 25^2 + 30^2 - 2 \cdot 25 \cdot 30 \cdot \cos(A) \][/tex]
Step-by-step solution:
1. Calculate the squares of the lengths:
[tex]\[ 20^2 = 400 \][/tex]
[tex]\[ 25^2 = 625 \][/tex]
[tex]\[ 30^2 = 900 \][/tex]
2. Substitute these values into the equation:
[tex]\[ 400 = 625 + 900 - 2 \cdot 25 \cdot 30 \cdot \cos(A) \][/tex]
3. Combine the constant terms on the right side:
[tex]\[ 400 = 1525 - 1500 \cos(A) \][/tex]
4. Rearrange to solve for \(\cos(A)\):
[tex]\[ 400 = 1525 - 1500 \cos(A) \][/tex]
[tex]\[ 400 - 1525 = -1500 \cos(A) \][/tex]
[tex]\[ -1125 = -1500 \cos(A) \][/tex]
5. Divide both sides by -1500:
[tex]\[ \cos(A) = \frac{-1125}{-1500} \][/tex]
[tex]\[ \cos(A) = 0.75 \][/tex]
Now, we need to find the angle \(A\) by taking the arccosine (inverse cosine) of 0.75.
[tex]\[ A = \arccos(0.75) \][/tex]
This angle in radians:
[tex]\[ A \approx 0.7227 \text{ radians} \][/tex]
Convert this angle to degrees:
[tex]\[ A \approx 41.41 \text{ degrees} \][/tex]
Approximately to the nearest degree:
[tex]\[ A \approx 41^\circ \][/tex]
Thus, the measure of the angle formed from the line from each lamppost to Bob is [tex]\( \boxed{41} \)[/tex] degrees.
