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=27.90 ft. x equals 27.90 ft. x=9.18 ft. x equals 9.18 ft. x=52.87 ft. x equals 52.87 ft. x=13.11 ft.

Answer 1

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

- The length of the ladder \( (hypotenuse) = 16 \) ft.
- The angle of elevation \( \theta = 55^\circ \).

The cosine of the angle is given by the formula:

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

In this case, the adjacent side is \( x \), and the hypotenuse is 16 ft. Thus, we have:

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

Now, rearranging this equation, we can solve for \( x \):

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

Next, we need to calculate \( \cos(55^\circ) \):

Using a calculator,

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

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

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

Rounding this to the nearest hundredth, we find:

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

So, the answer is:

**x = 9.18 ft.**