Question
A right triangle is formed by a ladder resting against a wall. The ladder is 20 feet long, and the top of the ladder rests 19 ft up the height of the wall. Find the approximate angle formed where the top of the ladder rests against the wall.(1 point)
Responses
43.5°
43.5 degrees
71.8°
71.8 degrees
18.2°
18.2 degrees
90°
Responses
43.5°
43.5 degrees
71.8°
71.8 degrees
18.2°
18.2 degrees
90°
Answers
GPT-4o mini
To find the angle formed where the top of the ladder rests against the wall, we can use the trigonometric function sine.
In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Here, the opposite side is the height the ladder reaches on the wall (19 ft), and the hypotenuse is the length of the ladder (20 ft).
The formula for sine is:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Substituting in the values we have:
\[
\sin(\theta) = \frac{19}{20}
\]
Now, to find \(\theta\), we take the inverse sine (arcsine) of the ratio:
\[
\theta = \sin^{-1}\left(\frac{19}{20}\right)
\]
Calculating this:
\[
\theta ≈ \sin^{-1}(0.95) \approx 71.8°
\]
Thus, the approximate angle formed where the top of the ladder rests against the wall is **71.8°**.
In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Here, the opposite side is the height the ladder reaches on the wall (19 ft), and the hypotenuse is the length of the ladder (20 ft).
The formula for sine is:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Substituting in the values we have:
\[
\sin(\theta) = \frac{19}{20}
\]
Now, to find \(\theta\), we take the inverse sine (arcsine) of the ratio:
\[
\theta = \sin^{-1}\left(\frac{19}{20}\right)
\]
Calculating this:
\[
\theta ≈ \sin^{-1}(0.95) \approx 71.8°
\]
Thus, the approximate angle formed where the top of the ladder rests against the wall is **71.8°**.