Question

A 15-foot ladder leans against a wall so that the ladder's angle of elevation is 42°
42
°
. Find x, the distance from the base of the ladder to the building. (Hint: Draw it out!)(1 point)
Responses

x = 22.42 ft
x = 22.42 ft

x = 11.15 ft
x = 11.15 ft

x = 20.18 ft
x = 20.18 ft

x = 10.04 ft

Answer 1

To solve for the distance \( x \) from the base of the ladder to the wall (building), we can use trigonometric relationships. In this case, we can use the cosine function since we know the angle of elevation and the length of the ladder (which acts as the hypotenuse of a right triangle).

The cosine of the angle is given by the formula:

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

Where:
- \(\theta = 42^\circ\)
- The adjacent side is the distance \( x \) from the base of the ladder to the wall.
- The hypotenuse is the length of the ladder = 15 feet.

So we have:

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

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

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

Next, we will calculate \( \cos(42^\circ) \):

Using a calculator:

\[
\cos(42^\circ) \approx 0.6691
\]

Now, substitute this value back into the equation:

\[
x \approx 15 \cdot 0.6691 \approx 10.04 \text{ ft}
\]

Thus, the distance from the base of the ladder to the building is:

\[
\boxed{10.04 \text{ ft}}
\]

So the correct response is \( x = 10.04 \text{ ft} \).