At a distance of 100 feet, the angle of elevation from the horizontal ground to the top of a building is `42^(@)` .
The height of the building is

Options

A. 67 feet
B. 74 feet
C. 90 feet
D. 110 feet

Explanation

To find the height of the building based on the given angle of elevation and distance from the building, we can use trigonometric principles. Here’s a step-by-step solution:

### Step 1: Understand the problem
We are given:
- The distance from the observer to the building (base) = 100 feet
- The angle of elevation (θ) = 42 degrees

We need to find the height of the building (h).

### Step 2: Set up the trigonometric relationship
In a right triangle formed by the height of the building, the distance from the observer to the base of the building, and the line of sight to the top of the building, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Here, the opposite side is the height of the building (h), and the adjacent side is the distance from the observer to the building (100 feet).

### Step 3: Write the equation
Using the tangent function:
\[ \tan(42^\circ) = \frac{h}{100} \]

### Step 4: Solve for h
Rearranging the equation to solve for h gives:
\[ h = 100 \cdot \tan(42^\circ) \]

### Step 5: Calculate the value of h
Now we need to calculate \( \tan(42^\circ) \). Using a calculator:
\[ \tan(42^\circ) \approx 0.9004 \]
Now substituting this value back into the equation for h:
\[ h \approx 100 \cdot 0.9004 \approx 90.04 \text{ feet} \]

### Step 6: Round the answer
Since we are looking for a practical answer, we can round this to:
\[ h \approx 90 \text{ feet} \]

### Final Answer
The height of the building is approximately **90 feet**.

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