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=9.18 ft.
x equals 9.18 ft.

x=13.11 ft.
x equals 13.11 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 using the cosine function, we can apply the cosine formula.

In this scenario, the cosine of the angle of elevation relates the adjacent side (the distance \( x \)) to the hypotenuse (the length of the ladder) as follows:

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

Where:
- \( \theta = 55^\circ \)
- The hypotenuse (the ladder) = 16 feet
- The adjacent side = \( x \)

We can rearrange this formula to solve for \( x \):

\[
x = \text{hypotenuse} \times \cos(\theta)
\]

Substituting the known values:

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

Using a calculator to find \( \cos(55^\circ) \):

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

Now substituting this value back into the equation for \( x \):

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

Rounding to the nearest hundredth, we find:

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

Thus, the answer is:

**x = 9.18 ft.**