Question
Robert leans a \( 20- \) foot ladder against a wall. If the ladder reaches 19.2 feet up the wall, what angle does form with the wall? Round your answer to the nearest tenth of a degree.
Ask by Murray Ray. in the United States
Jan 07,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The angle between the ladder and the wall is approximately 16.0 degrees.
Solution
To determine the angle that the ladder forms with the wall, we can use trigonometric relationships in a right triangle.
**Given:**
- **Ladder length (hypotenuse):** 20 feet
- **Height up the wall (adjacent side):** 19.2 feet
**Objective:**
Find the angle \( \theta \) between the ladder and the wall.
**Using the cosine function:**
\[
\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{19.2}{20} = 0.96
\]
To find \( \theta \), take the inverse cosine (arccos) of 0.96:
\[
\theta = \cos^{-1}(0.96)
\]
Calculating this:
\[
\theta \approx 16.0^\circ
\]
**Conclusion:**
The ladder forms an angle of **16.0 degrees** with the wall when rounded to the nearest tenth of a degree.
**Answer:**
The angle is approximately 16.0 degrees.
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Simplify this solution Mind Expander
To find the angle that the ladder forms with the wall, we can use the cosine function from trigonometry. Here, the adjacent side is the height the ladder reaches on the wall (19.2 feet), and the hypotenuse is the length of the ladder (20 feet). So, we can use the formula: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{19.2}{20} \] Calculating that gives us: \[ \cos(\theta) = 0.96 \] Now, we take the inverse cosine (arccos) to determine the angle: \[ \theta = \cos^{-1}(0.96) \approx 16.3^\circ \] So, the angle formed with the wall is approximately \(16.3\) degrees when rounded to the nearest tenth. The answer is \(16.3^\circ\).
