Problem 76 - Algebra and Trigonometry

Question

Two buildings of equal height are 800 feet apart. An observer on the street between the buildings measures the angles of elevation to the tops of the buildings as \(27^{\circ}\) and \(41^{\circ} .\) How high, to the nearest foot, are the buildings?

Step-by-Step Solution

Answer

The buildings are approximately 375 feet high each.

Step 1 - Identify Relevant Triangles

Visualize or draw the situation as two right-angle triangles sharing a common side. The side in common is the line from the observer to the bottom of the buildings, and the right angles are formed by this line and the lines extending from the observer to the tops of the buildings. The common side's length is half the distance between the two buildings, which is 400 feet.

Step 2 - Set Up Equations

Find the heights of the buildings by using the tangent of the angles, which equals the opposite (height) over the adjacent (400 feet). This gives us two equations: \(h = \tan(27^{\circ}) \times 400\) and \(h = \tan(41^{\circ}) \times 400\).

Step 3 - Solve Equations

After calculating, it's found that both equations yield equal values for 'h' since the two buildings are of equal height. By evaluating either of these expressions using a calculator, we get 'h' to be approximately 375 feet.