Question

a hot - air balloon is rising vertically. the angle of elevation from a point on level ground 123 feet from the balloon to a point directly under the passenger compartment changes from 17.5° to 30.3°. 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 in a right - triangle. $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let the initial height be $h_1$. Given $\theta_1 = 17.5^{\circ}$ and adjacent side $x = 123$ feet. So, $h_1=123\times\tan(17.5^{\circ})$.
$h_1 = 123\times0.3165=38.9295$ feet.

Step2: Find final height

Let the final height be $h_2$. Given $\theta_2 = 30.3^{\circ}$ and adjacent side $x = 123$ feet. Using $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, we have $h_2=123\times\tan(30.3^{\circ})$.
$h_2=123\times0.5844 = 71.8812$ feet.

Step3: Calculate the rise

The rise of the balloon $\Delta h=h_2 - h_1$.
$\Delta h=71.8812−38.9295 = 32.9517\approx33.0$ feet.

Answer:

$33.0$ feet