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) Responses x=14.39 ft. x equals 14.39 ft. x=28.79 ft. x equals 28.79 ft. x=13.89 ft. x equals 13.89 ft. x=43.23 ft.
Answers
GPT-4o mini
To find the distance \( x \) from the base of the ladder to the wall, we can use basic trigonometry. The ladder, the wall, and the ground form a right triangle, where:
- The length of the ladder is the hypotenuse (20 feet).
- The angle of elevation from the ground to the ladder is 46°.
- The distance \( x \) is the adjacent side of the triangle.
Using the cosine function:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
We can substitute in our values:
\[
\cos(46^\circ) = \frac{x}{20}
\]
To find \( x \):
\[
x = 20 \cdot \cos(46^\circ)
\]
Now, let's calculate \( \cos(46^\circ) \):
\[
\cos(46^\circ) \approx 0.6947
\]
So,
\[
x \approx 20 \cdot 0.6947 \approx 13.894
\]
Thus, rounding to two decimal places,
\[
x \approx 13.89 \text{ ft}
\]
The answer is \( x = 13.89 \, \text{ft} \).
- The length of the ladder is the hypotenuse (20 feet).
- The angle of elevation from the ground to the ladder is 46°.
- The distance \( x \) is the adjacent side of the triangle.
Using the cosine function:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
We can substitute in our values:
\[
\cos(46^\circ) = \frac{x}{20}
\]
To find \( x \):
\[
x = 20 \cdot \cos(46^\circ)
\]
Now, let's calculate \( \cos(46^\circ) \):
\[
\cos(46^\circ) \approx 0.6947
\]
So,
\[
x \approx 20 \cdot 0.6947 \approx 13.894
\]
Thus, rounding to two decimal places,
\[
x \approx 13.89 \text{ ft}
\]
The answer is \( x = 13.89 \, \text{ft} \).