A lifeguard is looking down at a swimmer in a swimming pool.

His sight forms an angle of depression of 55 degrees.

His seat is 10 ft up.

How far is the swimmer from the base of the lifeguard seat?

......ft [3]

[Total: 3]

To find the distance between the swimmer and the base of the lifeguard seat, we can use trigonometry. Specifically, we can apply the tangent function, which relates the angle of depression to the opposite side (the height of the lifeguard seat) and the adjacent side (the horizontal distance from the base of the seat to the swimmer).

Given:
- The height of the lifeguard seat = 10 ft
- The angle of depression = 55 degrees

Let \( x \) be the distance from the swimmer to the base of the lifeguard seat.

Using the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{10}{x}
\]
\[
\tan(55^\circ) = \frac{10}{x}
\]

Now, solve for \( x \):
\[
x = \frac{10}{\tan(55^\circ)}
\]

Using a calculator to find \(\tan(55^\circ)\):
\[
\tan(55^\circ) \approx 1.4281
\]

\[
x = \frac{10}{1.4281} \approx 7.0 \text{ ft}
\]

So, the swimmer is approximately **7.0 ft** from the base of the lifeguard seat.

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Would you like more details on this solution or have any other questions?

Here are some related questions:
1. How would the distance change if the lifeguard's seat height increased?
2. What if the angle of depression was 30 degrees instead of 55 degrees?
3. How do we calculate the swimmer's distance if the lifeguard was standing at the edge of the pool, not elevated?
4. What is the relationship between the angle of elevation and the angle of depression in this context?
5. How would you calculate the distance if only the horizontal distance was known, and the height and angle were unknown?

**Tip:** In trigonometry, the tangent function is especially useful for solving problems

Learn how to calculate the distance from a lifeguard's seat to a swimmer using the angle of depression and trigonometric principles. This problem involves applying the tangent function to find the horizontal distance given the height and angle of depression. Suitable for high school students.

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