Final answer:
To find the horizontal distance between the lifeguard and the person in distress, we can use trigonometry. The angle of depression is the angle between the lifeguard's line of sight and the horizontal. Using the tangent function, we can determine that the horizontal distance between the lifeguard and the person is approximately 22 feet.
Step-by-step explanation:
To find the horizontal distance between the lifeguard and the person in distress, we can use trigonometry. The angle of depression is the angle between the lifeguard's line of sight and the horizontal. Since we know the height of the lifeguard above the ground and the angle of depression, we can use the tangent function to find the horizontal distance.
- First, we need to find the length of the adjacent side (the horizontal distance) using the tangent function: tangent(angle) = opposite/adjacent.
- Substituting the values we know: tan(34°) = opposite/adjacent. In this case, the opposite side is the height of the lifeguard above the ground, which is 15 feet.
- Solving for the adjacent side (the horizontal distance): tan(34°) = 15/adjacent.
- Rearranging the equation to solve for adjacent, we get: adjacent = 15/tan(34°).
- Using a calculator, we can find the value of tan(34°) to be approximately 0.67101.
- Substituting this value into the equation: adjacent = 15/0.67101.
- Calculating the value of adjacent, we get adjacent ≈ 22.37229 feet.
Therefore, the horizontal distance between the lifeguard and the person is approximately 22 feet.