What value of 'p' will satisfy the equation `(p)/(sqrt(540))=(sqrt(240))/(p)` ?

Options

A. `6`
B. `10`
C. `6sqrt(10)`
D. `sqrt(10)`

Answer

To solve the equation \(\frac{p}{\sqrt{540}} = \frac{\sqrt{240}}{p}\), we can follow these steps:

### Step 1: Cross-multiply the equation
We start by cross-multiplying to eliminate the fractions:
\[ p^2 = \sqrt{240} \cdot \sqrt{540} \]

### Step 2: Simplify the right-hand side
Using the property of square roots, we can combine the square roots:
\[ p^2 = \sqrt{240 \cdot 540} \]

### Step 3: Calculate \(240 \cdot 540\)
Next, we need to calculate the product \(240 \cdot 540\):
\[ 240 \cdot 540 = 129600 \]

### Step 4: Take the square root
Now we take the square root of both sides:
\[ p^2 = \sqrt{129600} \]

### Step 5: Calculate \(\sqrt{129600}\)
Calculating the square root gives us:
\[ \sqrt{129600} = 360 \]

### Step 6: Solve for \(p\)
Now, we take the square root of both sides to find \(p\):
\[ p = \sqrt{360} \]

### Step 7: Simplify \(\sqrt{360}\)
We can simplify \(\sqrt{360}\):
\[ \sqrt{360} = \sqrt{36 \cdot 10} = 6\sqrt{10} \]

### Final Answer
Thus, the value of \(p\) that satisfies the equation is:
\[ p = 6\sqrt{10} \]

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