Problem 81 - Algebra and Trigonometry Real Mathematics, Real People

Question

In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is \(3.5^{\circ} .\) After you drive 13 miles closer to the mountain, the angle of elevation is \(9^{\circ}\) (see figure). Approximate the height of the mountain.

Step-by-Step Solution

Answer

To find the approximate height of the mountain, one must substitute the calculated value of 'd' into the first equation using a scientific calculator. The answer is the numerical value of 'h' approximated to sensible significant figures.

Step 1: Set up the observations as equations

The tangent of an angle in a right triangle is equal to the ratio of the opposite side to the adjacent side. Initially, let 'h' represent the height of the mountain and 'd' represent the distance to the mountain. Thus, we write the equation \(\tan(3.5^{\circ}) = \frac{h}{d}\). After driving 13 miles closer, the angle of elevation becomes \(9^{\circ}\), giving us \(\tan(9^{\circ}) = \frac{h}{d - 13}\).

Step 2: Solve the equations

Since both equations are equal to 'h', we can write \(\tan(3.5^{\circ})d = \tan(9^{\circ})(d - 13)\). Solving this equation for 'd' gives \(d = \frac{13\tan(9^{\circ})}{\tan(3.5^{\circ}) - \tan(9^{\circ})}\).

Step 3: Calculate mountain height

Substitute 'd' into the first equation to get \(h = \tan(3.5^{\circ}) \times \frac{13\tan(9^{\circ})}{\tan(3.5^{\circ}) - \tan(9^{\circ})}\). Then, use a scientific calculator to get an approximate value for 'h'.