To find the angle of elevation from the base of the stairs to the bottom of the door, we can use the inverse tangent (arctan) function. The height of the stairs is the opposite side of the triangle, and the distance out from the wall (the base of the triangle) is the adjacent side.
Given:
- Height (opposite side) = 3 feet
- Distance out from the wall (adjacent side) = 5 feet
We can find the angle \( \theta \) using the formula:
\[ \theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right) = \tan^{-1}\left(\frac{3}{5}\right) \]
Now, we can calculate that:
\[ \theta = \tan^{-1}\left(\frac{3}{5}\right) \]
Using a calculator, we find:
\[ \theta \approx 30.96 \text{ degrees} \]
Rounding this to the nearest whole degree, we get:
\[ \theta \approx 31 \text{ degrees} \]
Therefore, the angle of elevation from the base of the stairs to the bottom of the door is approximately 31 degrees.