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)

Question

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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 angle of elevation can be calculated using the formula:

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

In this case, the height is the vertical rise of the stairs (3 feet), and the base is the horizontal distance from the wall to the base of the stairs (5 feet).

Substituting the values into the formula:

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

Now we can calculate \(\frac{3}{5}\):

\[ \frac{3}{5} = 0.6 \]

Next, we find the arctangent of 0.6:

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

Using a calculator to find this value:

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

Rounding to the nearest whole degree:

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

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