Question

A 20-foot ladder leans against a wall so that the ladder's angle of elevation is 46°. Find x, the distance from the base of the ladder to the building. (1 point)
• x = 28.79 ft.
• x = 43.23 ft.
• x = 14.39 ft.
• x = 13.89 ft.

Answer 1

To find the distance \( x \) from the base of the ladder to the building, we can use the cosine function from the angle of elevation:

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

Here, the hypotenuse is the length of the ladder (20 feet) and the adjacent side is the distance from the base of the ladder to the building (x).

Given:
- \( \theta = 46^\circ \)
- Length of the ladder (hypotenuse) = 20 feet

Using the cosine function:

\[
\cos(46^\circ) = \frac{x}{20}
\]

Now, we can solve for \( x \):

\[
x = 20 \cdot \cos(46^\circ)
\]

Calculating \( \cos(46^\circ) \):

\[
\cos(46^\circ) \approx 0.6947
\]

Now, substituting the value back in:

\[
x \approx 20 \cdot 0.6947 \approx 13.894
\]

Thus,

\[
x \approx 13.89 \, \text{ft}
\]

The correct answer is:

**x = 13.89 ft.**