If `3sqrt(7)+sqrt(343)=19.21`, then find the value of `sqrt(252)+20sqrt(7)`.

Options

A. `39.4`
B. `49.9`
C. `56.8`
D. `92.3`

Answer

To solve the problem step-by-step, we will start from the given equation and find the required value.

### Step 1: Understand the given equation
We are given:
\[ 3\sqrt{7} + \sqrt{343} = 19.21 \]

### Step 2: Simplify \(\sqrt{343}\)
We know that:
\[ \sqrt{343} = \sqrt{7^3} = 7\sqrt{7} \]
So we can rewrite the equation as:
\[ 3\sqrt{7} + 7\sqrt{7} = 19.21 \]

### Step 3: Combine like terms
Now, combine the terms with \(\sqrt{7}\):
\[ (3 + 7)\sqrt{7} = 19.21 \]
This simplifies to:
\[ 10\sqrt{7} = 19.21 \]

### Step 4: Solve for \(\sqrt{7}\)
To find \(\sqrt{7}\), divide both sides by 10:
\[ \sqrt{7} = \frac{19.21}{10} = 1.921 \]

### Step 5: Find \(\sqrt{252}\)
Next, we need to find \(\sqrt{252}\). First, factor \(252\):
\[ 252 = 2 \times 126 = 2 \times 2 \times 63 = 2^2 \times 3^2 \times 7 \]
Thus,
\[ \sqrt{252} = \sqrt{2^2 \times 3^2 \times 7} = 2 \times 3 \times \sqrt{7} = 6\sqrt{7} \]

### Step 6: Calculate \(\sqrt{252} + 20\sqrt{7}\)
Now, we can substitute \(\sqrt{7}\) into the expression:
\[ \sqrt{252} + 20\sqrt{7} = 6\sqrt{7} + 20\sqrt{7} = (6 + 20)\sqrt{7} = 26\sqrt{7} \]

### Step 7: Substitute the value of \(\sqrt{7}\)
Now substitute \(\sqrt{7} = 1.921\):
\[ 26\sqrt{7} = 26 \times 1.921 \]

### Step 8: Calculate the final value
Now we compute:
\[ 26 \times 1.921 = 49.906 \]
Rounding this gives us approximately \(49.9\).

### Final Answer:
Thus, the value of \(\sqrt{252} + 20\sqrt{7}\) is approximately \(49.9\).

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