Question

caleb and emily are standing 100 yards from each other. caleb looks up at a 45° angle to see a hot air balloon. emily looks up at a 60° angle to see the same hot air balloon. approximately how far is the hot air balloon off the ground?
44.3 yd
63.4 yd
73.2 yd
89.7 yd

Explanation:

Step1: Let the height of the balloon be \( h \) yards and the distance of Caleb from the point directly below the balloon be \( x \) yards. Then the distance of Emily from that point is \( (100 - x) \) yards.

For Caleb: \( \tan45^{\circ}=\frac{h}{x}\), since \( \tan45^{\circ} = 1\), we have \( h=x\).
For Emily: \( \tan60^{\circ}=\frac{h}{100 - x}\), and \( \tan60^{\circ}=\sqrt{3}\), so \( h=\sqrt{3}(100 - x)\).

Step2: Substitute \( x = h \) into \( h=\sqrt{3}(100 - x)\)

We get \( h=\sqrt{3}(100 - h)\).
Expand: \( h = 100\sqrt{3}-\sqrt{3}h\).
Rearrange: \( h+\sqrt{3}h=100\sqrt{3}\).
Factor out \( h\): \( h(1 + \sqrt{3})=100\sqrt{3}\).
Solve for \( h\): \( h=\frac{100\sqrt{3}}{1+\sqrt{3}}\).
Rationalize the denominator: \( h=\frac{100\sqrt{3}(1 - \sqrt{3})}{(1+\sqrt{3})(1 - \sqrt{3})}=\frac{100\sqrt{3}- 300}{1 - 3}=\frac{300 - 100\sqrt{3}}{2}=150 - 50\sqrt{3}\approx150-50\times1.732 = 150 - 86.6=63.4\).

Answer:

63.4 yd