Problem 49 - Algebra and Trigonometry

Question

A hot-air balloon is rising vertically. From a point on level ground 125 feet from the point directly under the passenger compartment, the angle of elevation to the ballon changes from \(19.2^{\circ}\) to \(31.7^{\circ} .\) How far, to the nearest tenth of a foot, does the balloon rise during this period?

Step-by-Step Solution

Answer

To find the vertical distance the hot-air balloon rose, calculate the initial and final heights from the ground using the tangent of the angles given then take the difference between the final and initial height.

Step 1: Setup the problem

Draw the given scenario and depict it as a pair of right angle triangles. The observer's position forms the adjacent side (i.e. base) to the angle of elevation, the balloon's height makes up the opposite side (i.e. height), and the line of sight is the hypotenuse.

Step 2: Calculate the initial height of the balloon

Let the initial height of the balloon above the ground be \(h_1.\) From the tangent of the angle property of a right triangle, \(tan(19.2^{\circ}) = \frac{h_1}{125}\). Solving for \(h_1\) gives \(h_1 = 125 * tan(19.2^{\circ})\). Calculate this value.

Step 3: Calculate the final height of the balloon

Similarly, let the final height of the balloon above the ground be \(h_2.\) We have \(tan(31.7^{\circ}) = \frac{h_2}{125}\), giving \(h_2 = 125 * tan(31.7^{\circ}).\)

Step 4: Calculate the difference between final and initial height

The difference between the initial height and the final height of balloon i.e. how far the balloon rose can be calculated by taking the difference between \(h_2\) and \(h_1\), we get \( \Delta h = h_2 - h_1 \). Calculate using the values found in steps 2 and 3