Question

A 15-foot ladder leans against a wall so that the ladder's angle of elevation is 42°. Find x, the distance from the base of the ladder to the building. (Hint: Draw it out)

a. x = 11.15 ft
b. x = 22.42 ft
c. x = 10.04 ft
d. x = 20.18 ft

Answer 1

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

In this scenario, the ladder forms a right triangle with the wall and the ground. The ladder itself is the hypotenuse, the height up the wall is the side opposite the angle, and the distance from the base of the ladder to the wall is the adjacent side.

We can use the cosine of the angle of elevation:

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

Where:
- \( \theta = 42^\circ \)
- \(\text{hypotenuse} = 15 \, \text{ft}\)
- \(\text{adjacent} = x\)

So we have:

\[
\cos(42^\circ) = \frac{x}{15}
\]

To find \( x \), we can rearrange the equation:

\[
x = 15 \cdot \cos(42^\circ)
\]

Now we can calculate \( \cos(42^\circ) \):

Using a calculator:

\[
\cos(42^\circ) \approx 0.7431
\]

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

\[
x \approx 15 \cdot 0.7431 \approx 11.15 \, \text{ft}
\]

Therefore, the distance from the base of the ladder to the building is approximately:

\[
\boxed{11.15 \, \text{ft}}
\]

So the correct answer is (a) \( x = 11.15 \, \text{ft} \).