Problem 42 - Precalculus Essentials

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 is approximately 71 feet.

Step 1: Identify Given Information

The given information are as follows: angle of elevation \( \theta = 38.7^{\circ}\), distance from the building (adjacent side) = 30 yards.

Step 2: Apply Tangent Ratio of a Right Triangle

Knowing that \( \tan(\theta) = \frac{opposite}{adjacent}\), we substitute the values into the equation, so \( \tan(38.7^{\circ}) = \frac{height}{30 yards}\) where height is the opposite side we are trying to find.

Step 3: Solve for the Unknown

To solve for height, multiply both sides by 30 yards, therefore the equation becomes: height = 30 yards * \( \tan(38.7^{\circ})\).

Step 4: Calculate Results

Using a calculator, compute the product which gives approximately 23.77 yards.

Step 5: Convert Yards to Feet

We know that 1 yard is equivalent to 3 feet. Therefore, we have to multiply 23.77 yards by 3 to get the answer in feet. The solution, when rounded to the nearest foot, is approximately 71 feet.