Question

A 20-foot lader leans aganist a wall so that the ladder's angle of elevation is 46°. Find a, the distance from the base of the ladder to the building. (10 points)
Responses

x=28.79 ft.
x=28.79 ft.

x= 43.23 ft.
x= 43.23 ft.

x=14.39 ft.
x=14.39 ft.

x= 13.89 ft.

Answer 1

To find the distance from the base of the ladder to the building, we can use trigonometry. The scenario can be visualized as a right triangle where:

- The ladder is the hypotenuse (20 feet),
- The angle of elevation from the base of the ladder to the wall is 46°,
- The distance from the base of the ladder to the wall is \(x\), and
- The height at which the ladder touches the wall can be found using the sine of the angle.

We can use the cosine function, as it relates the adjacent side (the distance \(x\) from the base to the wall) to the hypotenuse (the length of the ladder).

The cosine function is defined as:

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

Substituting our values:

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

To isolate \(x\), we multiply both sides by 20:

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

Now, calculating \( \cos(46^\circ) \):

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

So, now we can find \(x\):

\[
x \approx 20 \cdot 0.6947 \approx 13.89 \, \text{ft}
\]

Thus, the distance from the base of the ladder to the building is approximately \( 13.89 \) feet.

So, the correct response is:

**x = 13.89 ft.**