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

Answer 1

To find the distance \( x \) between the foot of the ladder and the base of the building, we can use the cosine function from trigonometry.

Given:

• The length of the ladder (hypotenuse) \( h = 16 \) feet
• The angle of elevation \( \theta = 55^\circ \)

Using the cosine function:

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

In this situation, \( x \) (the distance from the foot of the ladder to the wall) is the adjacent side, and the hypotenuse is the length of the ladder.

So we can write the equation:

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

Now, solve for \( x \):

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

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

Using a calculator, we find:

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

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

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

Rounding \( x \) to the nearest hundredth:

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

The answer is:

\[ \boxed{9.18 \text{ ft}} \]

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