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

Answer 1

To find the distance \( x \) between the foot of the ladder and the base of the building using cosine, we can use the following cosine formula in a right triangle:

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

In this case, the angle \( \theta \) is 55°, the adjacent side is the distance \( x \) we want to find, and the hypotenuse is the length of the ladder (16 feet).

So we can write:

\[ \cos(55°) = \frac{x}{16} \]

Now, solving for \( x \):

\[ x = 16 \cdot \cos(55°) \]

Next, we can calculate \( \cos(55°) \):

\[ \cos(55°) \approx 0.5736 \]

Now, substitute this value into the equation:

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

Rounding to the nearest hundredth:

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

Thus, the correct response is \( x = 9.18 , \text{ft} \).