Question

charlie is watching hot - air balloons. balloon a has risen at a 56° angle. balloon b has risen at an 81° angle. if the distance from balloon a to the ground is 1210 feet, how far is balloon b from balloon a? round your answer to the nearest whole number.

Explanation:

Step1: Find the horizontal distance from Charlie to balloon A

We know that for balloon A, if the height of balloon A from the ground $h = 1210$ feet and the angle of elevation $x=56^{\circ}$. Using the tangent - function $\tan x=\frac{h}{d}$, where $h$ is the height and $d$ is the horizontal distance. So, $d=\frac{h}{\tan x}$.
$d=\frac{1210}{\tan56^{\circ}}\approx\frac{1210}{1.4826}\approx816.12$ feet.

Step2: Find the height of balloon B

Let the height of balloon B be $H$. Since the horizontal distance from Charlie to both balloons is the same (assuming they are on the same horizontal line relative to Charlie), and the angle of elevation of balloon B is $y = 81^{\circ}$. Using the tangent - function $\tan y=\frac{H}{d}$.
$H = d\times\tan81^{\circ}\approx816.12\times6.3138\approx5154.79$ feet.

Step3: Use the distance formula between two points in 2 - D (vertical distance between the two balloons)

The vertical distance between the two balloons is $\Delta h=H - 1210=5154.79 - 1210 = 3944.79$ feet. The horizontal distance between them is 0 (same horizontal position relative to Charlie). So the distance $D$ between balloon A and balloon B is the vertical distance between them.

Answer:

$3945$ feet