Question

a hot - air balloon is rising vertically. the angle of elevation from a point on level ground 125 feet from the balloon to a point directly under the passenger compartment changes from 19.4° to 33.3°. 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

We use the tangent function $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let the initial height be $h_1$. The adjacent side to the angle of elevation is 125 feet and the angle $\theta_1 = 19.4^{\circ}$. So $h_1=125\times\tan(19.4^{\circ})$. Calculating, $h_1 = 125\times0.3530=44.1250$ feet.

Step2: Find final height

Let the final height be $h_2$. The angle of elevation is $\theta_2 = 33.3^{\circ}$ and the adjacent side is still 125 feet. Using the tangent - function again, $h_2 = 125\times\tan(33.3^{\circ})$. Calculating, $h_2=125\times0.6584 = 82.3000$ feet.

Step3: Calculate the rise

The rise of the balloon is $\Delta h=h_2 - h_1$. Substituting the values of $h_1$ and $h_2$, we get $\Delta h=82.3000 - 44.1250=38.1750\approx38.2$ feet.

Answer:

$38.2$