What is the altitude of the air balloon to the nearest 100ft given the information below ?

A hot air balloon is floating above a straight stretch of highway. To estimate how high above the ground the balloon is floating, the passengers take measurement of a car below them. They assume that the car is traveling at 50mph. One minute after the car passes directly below the balloon they take a bearing on the car and find that the angle of depression of the car is 25 degrees.

Feb 24, 2018

$\approx 2000$ feet.

Explanation:

Firstly, creating a diagram with the given and needed information helps here:

Our first task is to find the distance, $d$, that the car has traveled since passing the balloon (accurately rendered in the above diagram).

We know that the car is traveling at 50 miles per hour, but we need to find the speed in hours per minute. We can do this by dividing 50 by 60:

$\frac{50 m i l \setminus e s}{1 h o u r} = \frac{50 m i l \setminus e s}{60 m i \setminus n \setminus u t e s} = .8 \overline{33}$ miles/minute

Since it's been one minute since the car passed the balloon, $d \cong .83$ miles.

Now to find $h$, the height:

$\tan \theta = \frac{o p p o s i t e}{a \mathrm{dj} a c e n t}$

$\tan 25 = \frac{h}{.83}$

$\tan 25 \cdot .83 \cong .387 \to h \cong .387$ miles.

There are 5,280 feet in a mile, so

${h}_{i \setminus n f e e t} \cong .387 \cdot 5280 \cong 2043.55 \cong 2000$ feet.

Hope this helps.