Question

there are many ropes keeping a hot air balloon from floating away before a balloon race. one of these ropes is fixed to the ground at a 45° angle. another is fixed to the ground at a 30° angle. if the hot air balloon is 21 feet off the ground, what is the distance between the ground directly underneath the balloon and the second rope rounded to the nearest hundredth of a foot? 12.12 14.85 29.70 36.37

Explanation:

Step1: Find the distance for the \(45^{\circ}\) angle

For the right - triangle with a \(45^{\circ}\) angle, \(\tan45^{\circ}=\frac{21}{x_1}\). Since \(\tan45^{\circ} = 1\), then \(x_1=21\) feet.

Step2: Find the distance for the \(30^{\circ}\) angle

For the right - triangle with a \(30^{\circ}\) angle, \(\tan30^{\circ}=\frac{21}{x_2}\). Since \(\tan30^{\circ}=\frac{1}{\sqrt{3}}\), then \(x_2 = 21\sqrt{3}\approx21\times1.732 = 36.37\) feet.

Step3: Calculate the total distance \(x\)

\(x=x_2 - x_1\). Substitute \(x_1 = 21\) and \(x_2=36.37\), we get \(x=36.37 - 21=14.85\) feet.

Answer:

14.85