A hot air balloon at an altitude of 2800 feet is 64 degrees. What is the new angle of elevation if the balloon decends veritally 1200 feet?

Question

This is from my per-calculus course.

Answer 1

Let's use the first sentence to find the distance from the point on the ground below the hot air balloon to the observer. You have a right triangle with one side 2800 feet and one side x feet. The angle between the x-foot side and the hypotenuse is 64 degrees. So, use tangent:

 

tan(64) = opposite/adjacent

tan(64) = 2800/x

x(tan(64)) = 2800

x = 2800/(tan(64))

x ≈ 1365.6513

 

 

 

Once the balloon descends by 1200 feet, it is at a height of 2800 - 1200 = 1600 feet. You'd like to know the new angle of elevation. Call that angle y. You have a right triangle with one side 1600 feet and one side 1365.6513 feet. The angle between the 1365.6513-foot side and the hypotenuse is y degrees. So, use tangent:

 

tan(y) = opposite/adjacent

tan(y) = 1600/1365.6513

tan(y) ≈ 1.1716

y ≈ tan^-1(1.1716)

y ≈ 49.5182

 

So, the new angle of elevation is approximately 49.5 degrees.