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.
A. x = 13.11 ft.
B. x = 27.90 ft.
C. x = 9.18 ft
D. x = 52.87 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 use the following relationship from trigonometry:

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

In our case:

- \(\theta = 55^\circ\) (the angle of elevation)
- The hypotenuse is the length of the ladder, which is 16 feet.
- The adjacent side (which we are trying to find) is \( x \).

Substituting into the cosine formula gives us:

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

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

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

Now, we can calculate \( \cos(55^\circ) \):

Using a calculator, we find:

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

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

\[
x = 16 \cdot 0.5736 \approx 9.177
\]

Rounding \( 9.177 \) to the nearest hundredth gives us:

\[
x \approx 9.18
\]

Thus, the distance \( x \) between the foot of the ladder and the base of the building is approximately:

**C. \( x = 9.18 \) ft.**