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)

Question

Responses

x=13.11 ft.
x equals 13.11 ft.

x=27.90 ft.
x equals 27.90 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. The setup of the problem is a right triangle where:

• The hypotenuse is the length of the ladder, which is 16 feet.
• The angle of elevation is 55°.
• The adjacent side (the distance from the foot of the ladder to the wall) is what we are trying to find, \( x \).

The cosine function is defined as:

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

In this case:

\[ \cos(55°) = \frac{x}{16} \]

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

\[ x = 16 \cdot \cos(55°) \]

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

\[ \cos(55°) \approx 0.5736 \]

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

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

Rounding \( 9.1776 \) to the nearest hundredth, we get:

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

So the correct response is:

x = 9.18 ft.