To find the angle of elevation from Miles to the top of the street light, we can use the right triangle formed by Miles, the top of the street light, and the base of the street light.
In this triangle:
- The diagonal distance (line of sight) to the top of the street light is the hypotenuse, which is 12 feet.
- The distance from Miles to the base of the street light is one of the legs of the triangle, which is 8 feet.
- We want to find the angle of elevation (let's call it θ) which is the angle opposite to the height of the street light.
Using the sine function, which is defined as: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] We can express this as: \[ \sin(\theta) = \frac{h}{12} \] where \( h \) is the height of the street light. We can also use the cosine function: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{8}{12} \] Thus: \[ \cos(\theta) = \frac{2}{3} \]
We can find \( \theta \) using the cosine inverse (arccos): \[ \theta = \cos^{-1}(\frac{2}{3}) \]
Calculating this gives: \[ \theta \approx 48.19 \text{ degrees} \]
Therefore, the angle of elevation with which Miles is looking at the bird is approximately \( 48.19 \) degrees.
The correct answer is: 48.19 degrees