To find the angle of elevation, we can use the sine function in the context of a right triangle. In this situation:
- The opposite side is the height of the street light (which we will find using the Pythagorean theorem).
- The adjacent side is the distance Miles is standing from the base of the street light (11 feet).
- The hypotenuse is the line of sight to the top of the street light (15 feet).
First, we can find the height of the street light using the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
Where:
- \(a\) is the height of the street light (opposite side),
- \(b\) is the distance from the base (11 feet),
- \(c\) is the hypotenuse (15 feet).
Rearranging the equation:
\[ a^2 + 11^2 = 15^2 \] \[ a^2 + 121 = 225 \] \[ a^2 = 225 - 121 \] \[ a^2 = 104 \] \[ a = \sqrt{104} \approx 10.2 \text{ feet} \]
Now, to find the angle of elevation \(\theta\), we can use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} = \frac{10.2}{11} \]
Now, we find the angle \(\theta\):
\[ \theta = \tan^{-1}\left(\frac{10.2}{11}\right) \]
Calculating this gives:
\[ \theta \approx \tan^{-1}(0.9273) \approx 42.83 \text{ degrees} \]
Therefore, the angle of elevation with which Miles is looking at the bird is approximately 42.83 degrees.
So the correct response is:
42.83 degrees.