Question

A 16-foot ladder leans against the wall so that the ladders 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

Answer 1

To solve for the distance \( x \) between the foot of the ladder and the base of the building, we can use the cosine of the angle.

Given:
- The length of the ladder (hypotenuse) = 16 feet
- The angle of elevation \( \theta = 55^\circ \)

The cosine of an angle in a right triangle is defined as the adjacent side (in this case, \( x \)) over the hypotenuse:

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

Now, substituting the values we have:

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

To find \( x \), we can rearrange the equation:

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

Now, calculating \( \cos(55^\circ) \) (in degrees):

Using a calculator:

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

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

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

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 **9.18 feet**.