What value of 'p' will satify the equation `p/(sqrt(80)) = (sqrt(320))/(p) ?`

Options

A. a. 6
B. b. 10
C. c. `4sqrt(10)`
D. d. `sqrt(10)`

Answer

To solve the equation \( \frac{p}{\sqrt{80}} = \frac{\sqrt{320}}{p} \), we will follow these steps:

### Step 1: Cross Multiply
We start by cross multiplying the equation:
\[ p \cdot p = \sqrt{320} \cdot \sqrt{80} \]
This simplifies to:
\[ p^2 = \sqrt{320 \cdot 80} \]

### Step 2: Calculate the Product Inside the Square Root
Next, we need to calculate \( 320 \cdot 80 \):
\[ 320 \cdot 80 = 25600 \]
So, we have:
\[ p^2 = \sqrt{25600} \]

### Step 3: Simplify the Square Root
Now, we simplify \( \sqrt{25600} \):
\[ \sqrt{25600} = \sqrt{256 \cdot 100} = \sqrt{256} \cdot \sqrt{100} = 16 \cdot 10 = 160 \]
Thus, we have:
\[ p^2 = 160 \]

### Step 4: Take the Square Root
To find \( p \), we take the square root of both sides:
\[ p = \sqrt{160} \]

### Step 5: Simplify \( \sqrt{160} \)
Now, we simplify \( \sqrt{160} \):
\[ \sqrt{160} = \sqrt{16 \cdot 10} = \sqrt{16} \cdot \sqrt{10} = 4\sqrt{10} \]
Thus, the value of \( p \) is:
\[ p = 4\sqrt{10} \]

### Final Answer
The value of \( p \) that satisfies the equation is:
\[ \boxed{4\sqrt{10}} \]