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.

Balloon 21 feet in air held by two ropes forming two right triangles. Rope 1 forms a base angle of 45 degrees. Rope 2 forms a base angle of 30 degrees.

If the hot air balloon is 21 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?

12.12
14.85
29.70
36.37

Answer 1

To find the distance between the ground directly underneath the balloon and the second rope, you can use trigonometry. Let's consider Rope 2, which forms a 30-degree angle with the ground.

You have a right triangle where the height (h) represents the height of the balloon above the ground (21 feet), and you want to find the base of this triangle (the distance to the second rope). The 30-degree angle is between the height (21 feet) and the base you're looking for. You can use the trigonometric function tangent (tan) to calculate the base:

tan(30 degrees) = Opposite / Adjacent

In this case:

Opposite = height of the balloon = 21 feet

Adjacent = the distance to the second rope (the base)

So, you can rearrange the formula to solve for the base:

Adjacent = Opposite / tan(30 degrees)

Adjacent = 21 feet / tan(30 degrees)

Now, calculate:

Adjacent = 21 feet / tan(30 degrees) ≈ 21 feet / 0.5774 ≈ 36.37 feet (rounded to the nearest hundredth of a foot)

So, the distance between the ground directly underneath the balloon and the second rope is approximately 36.37 feet. The answer is 36.37 feet.