To solve the problem, we need to determine the width of the river using the angle of depression and Paula's height. Here's a step-by-step solution:
1. Understanding the Problem:
- Paula stands at the side of a river and sights the opposite bank with an angle of depression of 7°.
- Paula's height is 5.5 feet.
- We need to find the horizontal distance (width of the river), where the line of sight forms a right triangle with Paula's height and the width of the river.
2. Visualize the Problem:
- Imagine a right triangle where:
- The vertical side is Paula's height, which is 5.5 feet.
- The horizontal side is the width of the river, which we need to find.
- The angle of depression from the horizontal is 7°, which is the angle above the horizontal axis down to the opposite side of the river.
3. Using Trigonometry:
- The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
- Here, the opposite side is Paula's height (5.5 feet), and the angle we are working with is 7°.
4. Tangent Function:
- Set up the tangent function:
[tex]\[
\tan(7°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5.5}{\text{width of the river}}
\][/tex]
5. Solve for the Width of the River:
- Rearrange the equation to solve for the width of the river:
[tex]\[
\text{width of the river} = \frac{5.5}{\tan(7°)}
\][/tex]
6. Calculate Using a Calculator:
- Use a scientific calculator to find the tangent of 7°.
- After calculating the width of the river:
[tex]\[
\text{width of the river} \approx 44.79390535386026 \text{ feet}
\][/tex]
7. Round to the Nearest Foot:
- The width of the river, rounded to the nearest foot, is approximately 45 feet.
So, the width of the river is approximately 45 feet.
