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. 16 feet < < 55 x feet (1 point) ○=9.18 ○=52.87 13.11 27.60

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 use the following relationship in a right triangle:

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

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

Using the cosine function:
\[
\cos(55^\circ) = \frac{x}{16}
\]

To find \( x \), we can rearrange the equation:
\[
x = 16 \cdot \cos(55^\circ)
\]

Now, we calculate \( \cos(55^\circ) \):
\[
\cos(55^\circ) \approx 0.5736
\]

Next, plug this value into the equation for \( x \):
\[
x \approx 16 \cdot 0.5736 \approx 9.1776
\]

Rounding to the nearest hundredth:
\[
x \approx 9.18
\]

Thus, the distance between the foot of the ladder and the base of the building is approximately \( \boxed{9.18} \) feet.