According to the website for the Balloon Federation of America*, one of the comp

Question

According to the website for the Balloon Federation of America*, one of the competitive games for the 2014 US Nationals was the "minimum distance double drop". In this event, the balloonist makes a fly in above a marked area on the ground and then tosses out a bean-bag marker into the target. The catch is that the minimum height of the balloon is 1.5 km. That means that the balloon must NOT descend to an altitude less than 1.5 km. Consider this: A judge on the ground looked up to a balloon at its position for dropping the marker on the target. The judge took a line of sight to the top of the balloon and the angle of elevation was 52.5 degree The judge then back-tracked a distance of 168 m and then determined that the new angle of elevation to balloon (it had not changed altitude) was 41.2 degree. How could the judge use this information and some right triangle trigonometry to determine the height of the balloon? Was the height of the balloon within the height restrictions for the contest? Be sure to draw a diagram and carefully explain your work. *http://www.bfa.net

Explanation / Answer

Let ‘h’ be the height of the balloon and let ‘b’ is the distance of the person from the balloon with an angle of 52.5 degrees and 168+b is the distance when the angle of elevation is 41.2 degrees

Consider the small right angled triangle,

we have tan (52.5 degrees) = h /b

In the big right angled triangle,

tan( 41.2) = h / b+ 168

tan( 52.5) = 1.3032 and tan(41.2) = .8752

Plugging in the values,

we have 1.3032 = h/b and .8752 = h/ b + 168

Solving the first equation above , we have

h = 1.3032 b.

plugging in this value in the second equation,

.8754( b + 168) = h = 1.3032b.

Hence

1.3032 b = .8754 b + 147.0672

Hence, b = 147.0672 / .4278 = 343.7758 m

Therefore h = 1.3032 * 343.7758 = 448.008 m

Height of the balloon = 448.008 m

The minimum height of the balloon is 1.5 km.

Therefore the height go the balloon 448.008 m is not within the height restrictions for the contest.

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