As Juliet stands 13 ft off the ground on her balcony, she spots Romeo in the distance, standing at ground level. She uses her clinometer and finds that his position is 4° from the horizontal (that is, the angle of depression is 4°).To the nearest foot, how far away is Romeo from a spot on the ground directly below her balcony?
Question
As Juliet stands 13 ft off the ground on her balcony, she spots Romeo in the distance, standing at ground level. She uses her clinometer and finds that his position is 4° from the horizontal (that is, the angle of depression is 4°).To the nearest foot, how far away is Romeo from a spot on the ground directly below her balcony?
Solution
To solve this problem, we can use the tangent of the angle of depression, which is the opposite side (the height from the ground to the balcony) divided by the adjacent side (the distance from Romeo to the spot on the ground directly below the balcony).
The formula is:
tan(θ) = opposite / adjacent
We know that the opposite side is 13 ft (the height of the balcony) and the angle of depression is 4°. We want to find the adjacent side (the distance from Romeo to the spot on the ground directly below the balcony).
So, we rearrange the formula to solve for the adjacent side:
adjacent = opposite / tan(θ)
Substitute the given values into the formula:
adjacent = 13 ft / tan(4°)
Using a calculator, we find that tan(4°) is approximately 0.06992.
So, the adjacent side is approximately 186 ft.
Therefore, Romeo is about 186 ft away from a spot on the ground directly below Juliet's balcony.
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