The unit of three physical quantities x, y and z are `g cm^(2) s^(-5)`, `gs^(-1)` and `cm s^(-2)` respectively. The relation between x, y and z is

Options

A. `x=yz^(2)`
B. `x=y^(2)z`
C. `y^(2)=xz`
D. `z=x^(2)y`

Explanation

To find the relation between the physical quantities \( x \), \( y \), and \( z \) given their units, we will express \( x \) in terms of \( y \) and \( z \) using the method of dimensional analysis.

### Step 1: Write down the units of the quantities
- The unit of \( x \) is \( g \, cm^2 \, s^{-5} \)
- The unit of \( y \) is \( g \, s^{-1} \)
- The unit of \( z \) is \( cm \, s^{-2} \)

### Step 2: Assume a relationship
We assume that \( x \) can be expressed in terms of \( y \) and \( z \) as:
\[ x = y^a \cdot z^b \]
where \( a \) and \( b \) are the powers we need to determine.

### Step 3: Write the units for the right-hand side
Substituting the units of \( y \) and \( z \) into the equation:
\[ x = (g \, s^{-1})^a \cdot (cm \, s^{-2})^b \]
This simplifies to:
\[ x = g^a \cdot cm^b \cdot s^{-a - 2b} \]

### Step 4: Equate the units
Now we equate the units from both sides:
- For \( g \): On the left side, we have \( 1 \) (from \( g^{1} \)), and on the right side, we have \( a \). Therefore:
\[ 1 = a \quad \text{(1)} \]

- For \( cm \): On the left side, we have \( 2 \) (from \( cm^2 \)), and on the right side, we have \( b \). Therefore:
\[ 2 = b \quad \text{(2)} \]

- For \( s \): On the left side, we have \( -5 \) (from \( s^{-5} \)), and on the right side, we have \( -a - 2b \). Therefore:
\[ -5 = -a - 2b \quad \text{(3)} \]

### Step 5: Substitute values from equations (1) and (2) into equation (3)
Substituting \( a = 1 \) and \( b = 2 \) into equation (3):
\[ -5 = -1 - 2(2) \]
This simplifies to:
\[ -5 = -1 - 4 \]
\[ -5 = -5 \quad \text{(True)} \]

### Step 6: Write the final relationship
Since we have found \( a = 1 \) and \( b = 2 \), we can express \( x \) in terms of \( y \) and \( z \):
\[ x = y^1 \cdot z^2 \]
Thus, the final relation is:
\[ x = y \cdot z^2 \]

### Conclusion
The relation between \( x \), \( y \), and \( z \) is:
\[ x = y \cdot z^2 \]