Question

From a hot-air balloon, Brody measures a 39degrees
∘
angle of depression to a landmark that’s 532 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.

Answer 1

To determine the vertical distance of the hot-air balloon above the ground based on the angle of depression, we can use trigonometry. The angle of depression from the balloon to the landmark is given as 39 degrees, and the horizontal distance to the landmark is 532 feet.

In this scenario, we have a right triangle where:
- The angle of depression is 39 degrees,
- The horizontal distance (adjacent side) is 532 feet,
- The vertical distance (opposite side) is what we want to find.

Since the angle of depression corresponds to the angle of elevation from the landmark to the balloon, we can use the tangent function, which is defined as:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

In this case, we can set up our equation as follows:

\[
\tan(39^\circ) = \frac{\text{vertical distance}}{532}
\]

Let’s denote the vertical distance as \(h\). Rearranging the equation gives us:

\[
h = 532 \cdot \tan(39^\circ)
\]

Now we can calculate \(h\):

1. Find \(\tan(39^\circ)\):
- Using a calculator, \(\tan(39^\circ) \approx 0.8098\).

2. Now calculate \(h\):
\[
h = 532 \cdot 0.8098 \approx 431.81 \text{ feet}
\]

Rounding to the nearest hundredth, the vertical distance above the ground is approximately:

\[
\boxed{431.81 \text{ feet}}
\]