From a hot-air balloon, Justin measures a 34degrees ∘ angle of depression to a landmark that’s 1424 feet away, measuring horizontally. What’s the balloon’s vertical distance above the ground? Round your answer to the nearest hundredth of a foot if necessary.
Question
From a hot-air balloon, Justin measures a 34degrees ∘ angle of depression to a landmark that’s 1424 feet away, measuring horizontally. What’s the balloon’s vertical distance above the ground? Round your answer to the nearest hundredth of a foot if necessary.
Solution 1
To solve this problem, we can use the tangent of the angle of depression, which is the ratio of the opposite side (the height of the balloon) to the adjacent side (the horizontal distance to the landmark).
The formula for tangent is:
tan(θ) = opposite/adjacent
We know that the angle of depression (θ) is 34 degrees and the adjacent side (the horizontal distance to the landmark) is 1424 feet. We want to find the opposite side (the height of the balloon), which we'll call h.
So we can set up the equation:
tan(34) = h/1424
To solve for h, we multiply both sides by 1424:
h = 1424 * tan(34)
Using a calculator, we find that tan(34) is approximately 0.6745.
So:
h = 1424 * 0.6745
h ≈ 960.47 feet
So, the balloon is approximately 960.47 feet above the ground.
Solution 2
To solve this problem, we can use the tangent of the angle of depression, which is the ratio of the opposite side (the height of the balloon) to the adjacent side (the horizontal distance to the landmark).
The formula for tangent is:
tan(θ) = opposite/adjacent
We know that the angle of depression (θ) is 34 degrees and the adjacent side (the horizontal distance to the landmark) is 1424 feet. We want to find the opposite side (the height of the balloon), which we'll call h.
So we can set up the equation:
tan(34) = h/1424
To solve for h, we multiply both sides by 1424:
h = 1424 * tan(34)
Using a calculator, we find that tan(34) is approximately 0.6745.
So:
h = 1424 * 0.6745
h ≈ 960.47 feet
So, the balloon is approximately 960.47 feet above the ground.
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