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=13.89 ft.
x equals 13.89 ft.
x=28.79 ft.
x equals 28.79 ft.
x=43.23 ft.
Responses
x=14.39 ft.
x equals 14.39 ft.
x=13.89 ft.
x equals 13.89 ft.
x=28.79 ft.
x equals 28.79 ft.
x=43.23 ft.
Answers
GPT-4o mini
To find the distance \( x \) from the base of the ladder to the building, we can use trigonometric functions. Specifically, we will use the cosine function since we know the adjacent side (the distance \( x \)) and the hypotenuse (the length of the ladder).
The cosine function is given by:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
In this situation:
- \(\theta = 46^\circ\)
- The hypotenuse (length of the ladder) = 20 ft.
- The adjacent side (distance from the base of the ladder to the building) = \( x \)
Substituting these values into the formula, we have:
\[
\cos(46^\circ) = \frac{x}{20}
\]
Now, rearranging to solve for \( x \):
\[
x = 20 \cdot \cos(46^\circ)
\]
Next, we calculate \( \cos(46^\circ) \) using a calculator:
\[
\cos(46^\circ) \approx 0.6947
\]
Now multiplying:
\[
x \approx 20 \cdot 0.6947 \approx 13.894
\]
Thus, \( x \approx 13.89 \) ft.
So the correct answer is:
**x = 13.89 ft.**
The cosine function is given by:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
In this situation:
- \(\theta = 46^\circ\)
- The hypotenuse (length of the ladder) = 20 ft.
- The adjacent side (distance from the base of the ladder to the building) = \( x \)
Substituting these values into the formula, we have:
\[
\cos(46^\circ) = \frac{x}{20}
\]
Now, rearranging to solve for \( x \):
\[
x = 20 \cdot \cos(46^\circ)
\]
Next, we calculate \( \cos(46^\circ) \) using a calculator:
\[
\cos(46^\circ) \approx 0.6947
\]
Now multiplying:
\[
x \approx 20 \cdot 0.6947 \approx 13.894
\]
Thus, \( x \approx 13.89 \) ft.
So the correct answer is:
**x = 13.89 ft.**