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
(2 points)
x = 43.23 feet
x = 28.79 feet
x = 13.89 feet
x = 14.39 feet

Answer 1

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

Given:
- The length of the ladder (hypotenuse) = 20 feet
- The angle of elevation = 46°

We can use the cosine function for this situation, which relates the adjacent side (the distance from the base of the ladder to the building, \( x \)) to the hypotenuse (the length of the ladder):

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

Substituting in the values we have:

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

Rearranging to solve for \( x \):

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

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

\[
\cos(46^\circ) \approx 0.6947 \quad (\text{using a calculator})
\]

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

\[
x \approx 20 \cdot 0.6947 \approx 13.894 \text{ feet}
\]

Rounding this to two decimal places, we have:

\[
x \approx 13.89 \text{ feet}
\]

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