Question

A 16-foot ladder leans against a wall so that the ladder’s angle of elevation is 55°. 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.(1 point)
Responses

x=13.11 ft.
x equals 13.11 ft.

x=9.18 ft.
x equals 9.18 ft.

x=52.87 ft.
x equals 52.87 ft.

x=27.90 ft.

Answer 1

To find the distance \( x \) between the foot of the ladder and the base of the building, we can use the cosine function in a right triangle formed by the ladder, the wall, and the ground.

In this triangle:
- The hypotenuse is the length of the ladder, which is 16 feet.
- The angle of elevation is 55°.
- \( x \) (the distance from the foot of the ladder to the base of the building) is the adjacent side.

According to the cosine definition:

\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]

In our case, this means:

\[
\cos(55°) = \frac{x}{16}
\]

To solve for \( x \), we rearrange the equation:

\[
x = 16 \cdot \cos(55°)
\]

Using a calculator to find \( \cos(55°) \):

\[
\cos(55°) \approx 0.5736
\]

Now, substitute this value into the equation:

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

Rounding to the nearest hundredth gives:

\[
x \approx 9.18 \text{ ft}
\]

Therefore, the correct answer is:

**x = 9.18 ft.**