If `sqrt(45)+sqrt(20)=11.180` find the value of `sqrt(180)+4 sqrt(5)`.

Options

A. `22.360`
B. `24.595`
C. `20.124`
D. `17.888`

Answer

To solve the problem step by step, we will start from the given equation and work our way to find the value of \( \sqrt{180} + 4\sqrt{5} \).

### Step 1: Understand the Given Information
We are given:
\[ \sqrt{45} + \sqrt{20} = 11.180 \]

### Step 2: Simplify the Square Roots
We can simplify \( \sqrt{45} \) and \( \sqrt{20} \):
\[ \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5} \]
\[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \]

### Step 3: Combine the Simplified Roots
Now, we can combine the simplified square roots:
\[ \sqrt{45} + \sqrt{20} = 3\sqrt{5} + 2\sqrt{5} = (3 + 2)\sqrt{5} = 5\sqrt{5} \]

### Step 4: Relate to the Given Value
From the information provided, we know:
\[ 5\sqrt{5} = 11.180 \]
To find \( \sqrt{5} \), we can divide both sides by 5:
\[ \sqrt{5} = \frac{11.180}{5} = 2.236 \]

### Step 5: Find \( \sqrt{180} + 4\sqrt{5} \)
Next, we need to calculate \( \sqrt{180} + 4\sqrt{5} \):
First, simplify \( \sqrt{180} \):
\[ \sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \cdot \sqrt{5} = 6\sqrt{5} \]

Now, substitute \( \sqrt{5} \) into the expression:
\[ \sqrt{180} + 4\sqrt{5} = 6\sqrt{5} + 4\sqrt{5} = (6 + 4)\sqrt{5} = 10\sqrt{5} \]

### Step 6: Substitute \( \sqrt{5} \) into the Expression
Now, we substitute the value of \( \sqrt{5} \):
\[ 10\sqrt{5} = 10 \cdot 2.236 = 22.36 \]

### Final Answer
Thus, the value of \( \sqrt{180} + 4\sqrt{5} \) is:
\[ \boxed{22.36} \]