`sqrt(9355)=?`

Options

A. 35
B. 115
C. 25
D. 96

Explanation

To find the approximate square root of 9355, we can follow these steps:

### Step 1: Identify the nearest perfect squares
First, we need to identify the perfect squares that are closest to 9355. The perfect squares we can consider are:
- \(81^2 = 6561\)
- \(90^2 = 8100\)
- \(95^2 = 9025\)
- \(96^2 = 9216\)
- \(97^2 = 9409\)
- \(98^2 = 9604\)

From this, we see that \(96^2 = 9216\) and \(97^2 = 9409\) are the closest perfect squares.

### Step 2: Choose a starting point
Since \(9355\) is between \(96^2\) and \(97^2\), we can start with \(96\) as our initial approximation.

### Step 3: Use the method of averaging
To refine our approximation, we can use the formula:
\[ \text{New Approximation} = \frac{\text{Old Approximation} + \frac{N}{\text{Old Approximation}}}{2} \]
Where \(N = 9355\) and our old approximation is \(96\).

Calculating:
\[ \text{New Approximation} = \frac{96 + \frac{9355}{96}}{2} \]

### Step 4: Calculate \(\frac{9355}{96}\)
Calculating:
\[ \frac{9355}{96} \approx 97.365625 \]

### Step 5: Update the approximation
Now substituting back into our formula:
\[ \text{New Approximation} = \frac{96 + 97.365625}{2} \approx \frac{193.365625}{2} \approx 96.6828125 \]

### Step 6: Round to the nearest whole number
Since we are looking for an approximate value, we can round \(96.6828125\) to \(97\).

### Conclusion
Thus, the approximate square root of \(9355\) is about \(96\).