Question

A 16-foot ladder leans against a wall so that the ladder's angle of elevation is 55 degrees. Find x, the distance between the foot of the ladder and the base of the building. Use cosine to solve for the missing side and round your answer to the nearest hundredth.

Answer 1

To find the distance \( x \) between the foot of the ladder and the base of the building, we will use the cosine function. In the context of a right triangle, where the ladder serves as the hypotenuse, the angle of elevation is given as \( 55^\circ \), and can be represented as follows:

- Hypotenuse (ladder length) = 16 feet
- Angle of elevation = \( 55^\circ \)
- Adjacent side (distance \( x \)) = distance from the foot of the ladder to the wall

Using the cosine function, we have:

\[
\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}
\]

Substituting in the known values:

\[
\cos(55^\circ) = \frac{x}{16}
\]

To isolate \( x \), we can multiply both sides by 16:

\[
x = 16 \cdot \cos(55^\circ)
\]

Next, we need to calculate \( \cos(55^\circ) \). Using a calculator:

\[
\cos(55^\circ) \approx 0.5736
\]

Now substituting back to find \( x \):

\[
x \approx 16 \cdot 0.5736 \approx 9.1776
\]

Rounding \( x \) to the nearest hundredth gives:

\[
x \approx 9.18
\]

Thus, the distance between the foot of the ladder and the base of the building is approximately **9.18 feet**.