Question

Stairs are built using horizontal planks placed within a right triangle dimension. If the ratio of the base and height of the triangle portion are not measured properly, the staircase may be too steep. Imagine you are building a short staircase for your garage that consists of three steps. The height from the garage floor to the bottom of the door is three feet, and you plan to build the stairs five feet out from the wall. Using the inverse of tangent, what is the angle of elevation from the base of the stairs to the bottom of the door? Round your answer to the nearest whole degree.(1 point)

Answer 1

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**.