What approximate value should come in place of the question mark(?) in the following questions ? (You are not expected to calculate the exact value.)
`sqrt1000+sqrt350+sqrt20=?`

Options

A. 45
B. 55
C. 65
D. 50

Answer

To solve the expression \( \sqrt{1000} + \sqrt{350} + \sqrt{20} \) and find the approximate value, we can break down each square root into simpler components.

### Step 1: Simplify \( \sqrt{1000} \)
We can express \( 1000 \) as \( 100 \times 10 \):
\[ \sqrt{1000} = \sqrt{100 \times 10} = \sqrt{100} \times \sqrt{10} = 10 \sqrt{10} \]
Now, we need to approximate \( \sqrt{10} \). The square root of \( 9 \) is \( 3 \) and the square root of \( 16 \) is \( 4 \), so \( \sqrt{10} \) is approximately \( 3.16 \). Therefore:
\[ \sqrt{1000} \approx 10 \times 3.16 = 31.6 \]

### Step 2: Simplify \( \sqrt{350} \)
We can express \( 350 \) as \( 100 \times 3.5 \):
\[ \sqrt{350} = \sqrt{100 \times 3.5} = \sqrt{100} \times \sqrt{3.5} = 10 \sqrt{3.5} \]
Next, we approximate \( \sqrt{3.5} \). The square root of \( 3.5 \) is between \( \sqrt{3} \) (approximately \( 1.73 \)) and \( \sqrt{4} \) (which is \( 2 \)). A reasonable approximation for \( \sqrt{3.5} \) is about \( 1.87 \):
\[ \sqrt{350} \approx 10 \times 1.87 = 18.7 \]

### Step 3: Simplify \( \sqrt{20} \)
We can express \( 20 \) as \( 4 \times 5 \):
\[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2 \sqrt{5} \]
Next, we approximate \( \sqrt{5} \). The square root of \( 5 \) is between \( \sqrt{4} \) (which is \( 2 \)) and \( \sqrt{9} \) (which is \( 3 \)). A reasonable approximation for \( \sqrt{5} \) is about \( 2.24 \):
\[ \sqrt{20} \approx 2 \times 2.24 = 4.48 \]

### Step 4: Combine the approximations
Now we can add the approximate values together:
\[ \sqrt{1000} + \sqrt{350} + \sqrt{20} \approx 31.6 + 18.7 + 4.48 \]
Calculating this gives:
\[ 31.6 + 18.7 = 50.3 \]
\[ 50.3 + 4.48 \approx 54.78 \]

### Final Approximation
Thus, the approximate value of \( \sqrt{1000} + \sqrt{350} + \sqrt{20} \) is about \( 55 \).

### Conclusion
The answer is approximately \( 55 \).

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