Question

question 11 (105 04 mc) (05 04 mc) charlie is watching hot air balloons. balloon a has risen at a 56° angle. balloon b has risen at an 81° angle. if the distance from balloon a to the ground is 1,200 feet, how far is balloon b from balloon a? round your answer to the nearest whole number. balloon a balloon b charlie

Explanation:

Step1: Find the horizontal distance from Charlie to Balloon A

Since the height of Balloon A is \(h = 1200\) feet and the angle of elevation \(x=56^{\circ}\), using the right - triangle trigonometry (\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)), for Balloon A, if we assume the horizontal distance from Charlie to the point directly below Balloon A is \(d_A\), and the line - of - sight distance is \(l_A\). But since the height \(h = 1200\) feet and \(\tan x=\frac{h}{d_A}\), here \(x = 56^{\circ}\), and \(h\) (opposite side) and \(d_A\) (adjacent side) are related by \(\tan56^{\circ}=\frac{1200}{d_A}\), so \(d_A=\frac{1200}{\tan56^{\circ}}\approx\frac{1200}{1.4826}\approx810.74\) feet.

Step2: Find the horizontal distance from Charlie to Balloon B

For Balloon B, let the height be the same as Balloon A (assuming they are at the same vertical level, which is a reasonable assumption from the right - triangle setup in the diagram). The angle of depression \(y = 81^{\circ}\), and if the horizontal distance from Charlie to the point directly below Balloon B is \(d_B\), using \(\tan y=\frac{h}{d_B}\) (where \(h = 1200\) feet, opposite side, and \(d_B\) is adjacent side). So \(d_B=\frac{1200}{\tan81^{\circ}}\approx\frac{1200}{6.3138}\approx190.06\) feet.

Step3: Calculate the distance between Balloon A and Balloon B

The distance \(D\) between Balloon A and Balloon B is \(d_A + d_B\) (since they are on opposite sides of Charlie in the horizontal direction). \(D=\frac{1200}{\tan56^{\circ}}+\frac{1200}{\tan81^{\circ}}\approx810.74 + 190.06=1000.8\approx1001\) feet.

Answer:

\(1001\) feet