# 5. The Height of a Balloon

On a sunny day, tie a helium balloon to a long string. Ask the students, “How high, above the ground, is the balloon?”

Of course, this is a variation on the classic textbook “poles casting shadows” problems. Instead of a helium balloon, you can use a Chinese lantern, a vertically thrown ball with a string attached, … anything, as long as its height is empirically known. As in the Distance to an Unapproachable Point MathLab, this can be solved by either geometry (using similar triangles) or right triangle trigonometry. All that is required is to determine the elevation of the sun and then compare the “balloon right triangle” (vertices are the point on the ground directly below the balloon, the balloon itself, and the balloon’s shadow) with, say, the right triangle whose legs are a vertical meter stick and its shadow.

Clearly,

Q. The real world question is, “How high is that balloon above the ground?”

R. The mathematics required to resolve this question is either simple triangle similarity or right triangle trigonometry.

V. Of course, an actual measurement of the length of the attached string, from ground to balloon, provides the simple empirical verification of how well the students performed.