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 hundreth.
a. x = 27.90 ft.
b. x = 13.11 ft.
c. x = 52.87 ft.
d. x = 9.18 ft.

Answer 1

To solve for the distance \( x \) between the foot of the ladder and the base of the building using the cosine function, we can use the formula:

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

In this case, the angle \( \theta = 55^\circ \), the adjacent side is \( x \) (the distance from the wall to the foot of the ladder), and the hypotenuse is the length of the ladder, which is 16 feet.

Therefore, we can set up the equation as follows:

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

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

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

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

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

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

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

Rounding to the nearest hundredth, we get:

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

Thus, the correct answer is:

d. \( x = 9.18 \, \text{ft} \)