What approximate value should come in place of the question mark (?) in the following questions ? (Note : You are not expected to calculate the exact value)
`root2(4663) + 349 = ? + 21.003`

Options

A. 760
B. 465
C. 400
D. 350

Answer

To solve the equation \( \sqrt{4663} + 349 = ? + 21.003 \), we will follow these steps:

1. **Calculate \( \sqrt{4663} \)**:
- To estimate \( \sqrt{4663} \), we can find two perfect squares between which 4663 lies.
- \( 68^2 = 4624 \) and \( 69^2 = 4761 \). Therefore, \( \sqrt{4663} \) is approximately between 68 and 69.
- A rough estimate gives us \( \sqrt{4663} \approx 68.3 \).

2. **Add 349 to the approximate square root**:
- Now, we add 349 to our estimate of \( \sqrt{4663} \):
\[ 68.3 + 349 \approx 417.3 \]

3. **Set up the equation**:
- We now have:
\[ 417.3 \approx ? + 21.003 \]

4. **Isolate the question mark (?)**:
- To find the value of ?, we can rearrange the equation:
\[ ? = 417.3 - 21.003 \]

5. **Perform the subtraction**:
- Now, we calculate:
\[ 417.3 - 21.003 \approx 396.297 \]

6. **Round to the nearest approximate value**:
- The approximate value of \( ? \) is about 396.297, which we can round to 400 for simplicity.

7. **Choose the closest option**:
- From the given options (760, 465, 400, 350), the closest value to our calculated approximate value of 396.297 is 400.

Thus, the approximate value that should come in place of the question mark (?) is **400**.