Question

a hot - air balloon is rising vertically. the angle of elevation from a point on level ground 127 feet from the balloon to a point directly under the passenger compartment changes from 19.9° to 30.8°. how far, to the nearest tenth of a foot, does the balloon rise during this period? feet (round the final answer to one decimal place as needed. round all intermediate values to four decimal places as needed.)

Explanation:

Step1: Find initial height

Use the tangent function $\tan\theta=\frac{opposite}{adjacent}$. Let the initial height be $h_1$. Given $\theta_1 = 19.9^{\circ}$ and adjacent side $x = 127$ feet. So $h_1=127\times\tan(19.9^{\circ})$.
$h_1 = 127\times0.3639\approx46.1153$ feet.

Step2: Find final height

Let the final height be $h_2$. Given $\theta_2=30.8^{\circ}$ and adjacent side $x = 127$ feet. Using $\tan\theta=\frac{opposite}{adjacent}$, we have $h_2 = 127\times\tan(30.8^{\circ})$.
$h_2=127\times0.5919\approx75.1713$ feet.

Step3: Find the rise

The rise of the balloon is $h = h_2 - h_1$.
$h=75.1713 - 46.1153=29.056\approx29.1$ feet.

Answer:

$29.1$