Question

a hot - air balloon is flying at an altitude of 2784 feet. if the angle of depression from the pilot in the balloon to a house on the ground below is 27°, how far is the house from the pilot?
the house is (square) feet from the pilot of the balloon.
(do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Use the sine function

We know that the angle of depression is \(27^{\circ}\), and the altitude (opposite side) is \(h = 2784\) feet. Let the distance from the pilot to the house (hypotenuse) be \(d\). The sine of an angle in a right - triangle is given by \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Since the angle of depression is equal to the angle of elevation from the house to the balloon, \(\theta = 27^{\circ}\) and \(\sin(27^{\circ})=\frac{2784}{d}\).

Step2: Solve for \(d\)

We can rewrite the equation \(\sin(27^{\circ})=\frac{2784}{d}\) as \(d=\frac{2784}{\sin(27^{\circ})}\). We know that \(\sin(27^{\circ})\approx0.454\). Then \(d=\frac{2784}{0.454}\).

$$d=\frac{2784}{0.454}\approx6132.2$$

Answer:

\(6132.2\)