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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.

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Solution

To solve this problem, we can use the tangent of the angle of depression. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In this case, the angle is 34 degrees, the adjacent side is the horizontal distance to the landmark (1424 feet), and the opposite side is the vertical distance from the balloon to the ground, which we're trying to find.

The formula for tangent is:

tan(θ) = opposite/adjacent

We can rearrange this to solve for the opposite side:

opposite = tan(θ) * adjacent

Substituting the given values:

opposite = tan(34 degrees) * 1424 feet

Using a calculator, tan(34 degrees) is approximately 0.6745.

So, the vertical distance from the balloon to the ground is:

opposite = 0.6745 * 1424 feet = 960.408 feet

Rounding to the nearest hundredth of a foot, the balloon is approximately 960.41 feet above the ground.

This problem has been solved

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