Problem 77 - Algebra and Trigonometry

Question

The angle of elevation to the top of a building changes from \(20^{\circ}\) to \(40^{\circ}\) as an observer advances 75 feet toward the building. Find the height of the building to the nearest foot.

Step-by-Step Solution

Answer

The height of the building, rounded to the nearest foot, is obtained by substituting the actual degree values into the equation and solving it.

Step 1: Identify the right triangles

There are two right triangles involved in this problem. In the first triangle, the angle of elevation to the top of the building is \(20^{\circ}\), but after the observer moves closer to the building by 75 feet, the angle of elevation becomes \(40^{\circ}\). Taking the observer's line of sight as the hypotenuse, we can identify two right triangles. Let's denote the initial distance from the building as \(d\), the height of the building as \(h\), and the distance the observer moved towards the building as 75 feet.

Step 2: Setup the equations using tangent

Using the definition of tangent (opposite over adjacent) in both triangles, we can setup the equations: From the first triangle, \(\tan(20^{\circ}) = \frac{h}{d}\) and from the second triangle, \(\tan(40^{\circ})= \frac{h}{d-75}\).

Step 3: Solve the system of equations

Now we have a system of two equations with two variables, \(d\) and \(h\). We can now solve the system of equations, perhaps by using substitution or elimination method. Using substitution, we substitute \(\frac{h}{d}\) from the first equation into the second to get: \(\tan(40^{\circ})= \tan(20^{\circ}) \cdot \frac{d}{d-75}\). Solving for \(d\) gives \(d = \frac{75}{\tan(40^{\circ}) - \tan(20^{\circ})}\).

Step 4: Find the height of the building

Substitute the value of \(d\) into the first equation \(\tan(20^{\circ}) = \frac{h}{d}\) and solve for \(h\). It gives \(h = d \cdot \tan(20^{\circ})\). Substitute \(d\) into the equation to get \(h = 75 \cdot \frac{\tan(20^{\circ})}{\tan(40^{\circ}) - \tan(20^{\circ})}\).

Step 5: Compute the actual value

Now substitute the actual degrees into the equation and calculate \(h\), which gives the result in feet. The value needs to be rounded to the nearest foot according to the problem statement.