In an auditorium, the number of rows is equal to the number of chairs in each row. If the capacity of the auditorium is `2025`, find the number of chairs in each row.

Options

A. `45`
B. `35`
C. `25`
D. `15`

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Define the Variables
Let the number of rows in the auditorium be \( x \). Since the number of chairs in each row is also equal to the number of rows, the number of chairs in each row is also \( x \).

### Step 2: Set Up the Equation
The total capacity of the auditorium is given as \( 2025 \). Therefore, we can express the total capacity as:
\[ \text{Total Capacity} = \text{Number of Rows} \times \text{Number of Chairs per Row} = x \times x = x^2 \]
This gives us the equation:
\[ x^2 = 2025 \]

### Step 3: Solve for \( x \)
To find \( x \), we need to take the square root of both sides of the equation:
\[ x = \sqrt{2025} \]

### Step 4: Calculate the Square Root
Now we will calculate \( \sqrt{2025} \). We can do this using the long division method or by recognizing perfect squares.

1. **Long Division Method**:
- Pair the digits of \( 2025 \) from the right: \( 20 | 25 \).
- The largest square less than or equal to \( 20 \) is \( 4 \) (since \( 4^2 = 16 \)).
- Subtract \( 16 \) from \( 20 \) to get \( 4 \), and bring down the next pair \( 25 \) to make it \( 425 \).
- Double the quotient \( 4 \) to get \( 8 \) and find a digit \( d \) such that \( (80 + d) \cdot d \) is less than or equal to \( 425 \).
- Testing \( d = 5 \): \( 85 \cdot 5 = 425 \).
- This gives us \( 45 \) as the quotient.

Thus, \( \sqrt{2025} = 45 \).

### Step 5: Conclusion
The number of chairs in each row, which is \( x \), is:
\[ x = 45 \]

### Final Answer
The number of chairs in each row is **45**.

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