Problem 42 - Algebra and Trigonometry

Question

From a point on level ground 30 yards from the base of a building, the angle of elevation is \(38.7^{\circ} .\) Approximate the height of the building to the nearest foot.

Step-by-Step Solution

Answer

The height of the building, to the nearest foot, is given by the expression \[ h = 90 \times tan(38.7) \]

Step 1: Identify the Known Variables

The distance from the building at where the observation is made is 30 yards, which is equivalent to 90 feet (as one yard equals three feet). Also, the angle of elevation is \(38.7^{\circ}\). These form two sides of a right-angled triangle—we are looking for the height of the building, which is the side opposite the angle.

Step 2: Apply Tangent Function

The tangent function of an angle in a right triangle is defined as the ratio of the side opposite the angle to the side adjacent. Therefore, we can express the height of the building (which we'll call h) in relation to the distance and the angle as follows: \[ tan(38.7^{\circ}) = \frac{h}{90} \]

Step 3: Solve for h

Rearranging for h, we get: \[ h = 90 \times tan(38.7^{\circ}) \] Now all we have to do is calculate this expression to obtain the height of the building.

Step 4: Calculate and Round

Perform the calculation with precision. Afterward, approximate the result to the nearest foot, as instructed in the problem.