If `sqrt(32)+sqrt(72)=14.14` then `sqrt(18)+sqrt(50)+sqrt(98)+sqrt(1250)`=?

Options

A. 45.45
B. 56.56
C. 67.67
D. 78.78

Answer

To solve the problem step by step, we will first simplify the square roots and then calculate the final result.

### Step 1: Simplify the square roots
We need to express each square root in a simplified form.

1. **Calculate \( \sqrt{32} \)**:
\[ \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} \]

2. **Calculate \( \sqrt{72} \)**:
\[ \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} \]

### Step 2: Add the simplified square roots
Now we add the two results:
\[ \sqrt{32} + \sqrt{72} = 4\sqrt{2} + 6\sqrt{2} = (4 + 6)\sqrt{2} = 10\sqrt{2} \]

### Step 3: Set up the equation
According to the problem, we know:
\[ 10\sqrt{2} = 14.14 \]
From this, we can find the value of \( \sqrt{2} \):
\[ \sqrt{2} = \frac{14.14}{10} = 1.414 \]

### Step 4: Calculate the new expression
Now we need to calculate \( \sqrt{18} + \sqrt{50} + \sqrt{98} + \sqrt{1250} \).

1. **Calculate \( \sqrt{18} \)**:
\[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \]

2. **Calculate \( \sqrt{50} \)**:
\[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \]

3. **Calculate \( \sqrt{98} \)**:
\[ \sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \cdot \sqrt{2} = 7\sqrt{2} \]

4. **Calculate \( \sqrt{1250} \)**:
\[ \sqrt{1250} = \sqrt{25 \times 50} = \sqrt{25} \cdot \sqrt{50} = 5\sqrt{50} = 5\sqrt{25 \times 2} = 5 \cdot 5\sqrt{2} = 25\sqrt{2} \]

### Step 5: Combine all the square roots
Now we can add these results together:
\[ \sqrt{18} + \sqrt{50} + \sqrt{98} + \sqrt{1250} = 3\sqrt{2} + 5\sqrt{2} + 7\sqrt{2} + 25\sqrt{2} \]
\[ = (3 + 5 + 7 + 25)\sqrt{2} = 40\sqrt{2} \]

### Step 6: Substitute the value of \( \sqrt{2} \)
Now we substitute \( \sqrt{2} = 1.414 \):
\[ 40\sqrt{2} = 40 \times 1.414 = 56.56 \]

### Final Answer
Thus, the value of \( \sqrt{18} + \sqrt{50} + \sqrt{98} + \sqrt{1250} \) is:
\[ \boxed{56.56} \]