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
Answers
GPT-4o mini
To find the distance \( x \) from the base of the ladder to the building, we can use trigonometric functions. In this case, we will use the cosine function since we have the hypotenuse (the length of the ladder) and we want to find the adjacent side (the distance \( x \)).
Given:
- Length of the ladder (hypotenuse) = 20 feet
- Angle of elevation \( \theta = 46^\circ \)
The cosine of the angle can be expressed as:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
Substituting our values into this formula:
\[
\cos(46^\circ) = \frac{x}{20}
\]
To find \( x \):
\[
x = 20 \cdot \cos(46^\circ)
\]
Now we need to calculate \( \cos(46^\circ) \):
Using a calculator:
\[
\cos(46^\circ) \approx 0.6947
\]
Now substitute back to find \( x \):
\[
x \approx 20 \cdot 0.6947 \approx 13.894
\]
Rounding to two decimal places:
\[
x \approx 13.89 \text{ feet}
\]
Thus, the distance from the base of the ladder to the building is approximately **13.89 feet**.
Given:
- Length of the ladder (hypotenuse) = 20 feet
- Angle of elevation \( \theta = 46^\circ \)
The cosine of the angle can be expressed as:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
Substituting our values into this formula:
\[
\cos(46^\circ) = \frac{x}{20}
\]
To find \( x \):
\[
x = 20 \cdot \cos(46^\circ)
\]
Now we need to calculate \( \cos(46^\circ) \):
Using a calculator:
\[
\cos(46^\circ) \approx 0.6947
\]
Now substitute back to find \( x \):
\[
x \approx 20 \cdot 0.6947 \approx 13.894
\]
Rounding to two decimal places:
\[
x \approx 13.89 \text{ feet}
\]
Thus, the distance from the base of the ladder to the building is approximately **13.89 feet**.