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.

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 15 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?

 21.21 25.98 30 51.96

Answer 1

Let's call the distance from the balloon to the rope 2 anchor, L, and note that L is the hypotenuse of the larger right triangle. Using the definition of the sine function,

15 ft / L = sin30° = 0.5, which yields the equation L = 15 ft / 0.5 = 30 ft.

Pythagorus says: (15 ft)² + x² = (30 ft)², which leads to x = SQRT [30² - 15²] ft = SQRT [900 - 225] ft

= SQRT [675] ft

= 25.98 ft.

The step including 900 and 225 is unnecessary, but I included it because long ago, a math teacher had

my class memorize the squares of the numbers 13-25. This has saved me a great deal of calculation time over the years.

Answer 2

The triangle with the 30 degree acute angle is a 30-60-90 right triangle where the 15 feet is opposite 30 degrees. That means the 15 foot side is the short leg. In a 30-60-90 right triangle the long leg (in this case the leg from under the balloon to the location of rope 2) is √3 times the short leg.

So, 15√3 (get out your calculator)

Rope 1 has nothing to do with answering this question. Perhaps there are additional parts to the question, like distance from ground under balloon to rope 1 (now you have a 45,45 right triangle where legs are equal). What is the distance from rope1 to rope 2?