Stairs are built using horizontal planks placed within a right triangle dimension.

Question

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 using the inverse tangent function, we can use the formula:

\[ \theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right) \]

In this case, the "opposite" side is the height from the garage floor to the bottom of the door, which is 3 feet. The "adjacent" side is the distance from the wall to the base of the stairs, which is 5 feet.

Substituting in the values, we get:

\[ \theta = \tan^{-1}\left(\frac{3}{5}\right) \]

Now we calculate:

\[ \theta = \tan^{-1}(0.6) \]

Using a calculator, we find that:

\[ \theta \approx 30.96 \text{ degrees} \]

Rounding this to the nearest whole degree:

\[ \theta \approx 31 \text{ degrees} \]

Thus, the angle of elevation from the base of the stairs to the bottom of the door is approximately 31 degrees.