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How to Determine the Altitude of a Balloon

Trigonometry has many applications for finding distances. For example, to find the altitude of a floating object, you can use the angle between one sighting of the object and a second sighting to solve for its distance from the ground.

image0.jpg

Cindy and Mindy, standing a mile apart, spot a hot-air balloon directly above a particular point on the ground. The angle of elevation from Cindy to the balloon is 60 degrees; the angle of elevation from Mindy to the balloon is 70 degrees. The preceding figure shows a visual representation. How high is the balloon?

If you look at the figure, you see that two right triangles are formed. The two triangles share a side — the one opposite the measured acute angle in each. Call its length y. The two adjacent sides add up to 1 mile, so you can keep the variables to a minimum by naming one side x and the other 1–x.

image1.jpg

The preceding figure shows the triangles with the variables.

To figure out how high the balloon is, follow these steps:

  1. Identify the parts of the triangles that you can use to solve the problem.

    In both triangles, you have variables for the adjacent and opposite sides of the acute angles of elevation.

  2. Determine which trig function to use.

    The tangent uses the opposite and adjacent sides.

  3. Write equations with the trig functions.

    image2.png
  4. Solve for x by setting the equations equal to one another.

    Solve each of the equations for y.

    image3.png

    Set those two equations equal to one another and solve for x.

    image4.png
  5. Solve for the value of x.

    You find the value of x by finding the values of the functions with a calculator. Upon doing so, you find that x is approximately 0.613 miles.

    image5.png

    Put that value into one of the equations to solve for y:

    image6.png

    The balloon is 1.062 miles high — sounds a tad high!

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