The dimensions of `a/b` in the equation `P=(a-t^(2))/(bx)` where `P` is pressure, `x` is distance and `t` is time are

Options

A. `[M^(2)LT^(-3)]`
B. `[MT^(-2)]`
C. `[ML^(3)T^(-1)]`
D. `[LT^(-3)]`

Answer

To find the dimensions of \( \frac{a}{b} \) in the equation \( P = \frac{a - t^2}{bx} \), we will follow these steps:

### Step 1: Understand the given equation
We have the equation:
\[ P = \frac{a - t^2}{bx} \]
where \( P \) is pressure, \( x \) is distance, and \( t \) is time.

### Step 2: Identify the dimensions of pressure \( P \)
Pressure is defined as force per unit area. The dimensions of force are given by:
\[ \text{Force} = \text{mass} \times \text{acceleration} = M \cdot LT^{-2} = MLT^{-2} \]
Since pressure is force per unit area, we have:
\[ [P] = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \]

### Step 3: Identify the dimensions of \( t^2 \)
The dimension of time \( t \) is:
\[ [t] = T \]
Thus, the dimensions of \( t^2 \) are:
\[ [t^2] = T^2 \]

### Step 4: Determine the dimensions of \( a \)
From the equation, since \( a \) and \( t^2 \) are being subtracted, they must have the same dimensions. Therefore, the dimensions of \( a \) are:
\[ [a] = T^2 \]

### Step 5: Identify the dimensions of \( x \)
The dimension of distance \( x \) is:
\[ [x] = L \]

### Step 6: Rearranging the equation to find \( b \)
Rearranging the equation gives:
\[ bx = a - t^2 \]
This implies:
\[ b = \frac{a - t^2}{x} \]
Since \( a \) and \( t^2 \) have the same dimensions, we can express this as:
\[ b = \frac{T^2}{L} \]

### Step 7: Determine the dimensions of \( b \)
Thus, the dimensions of \( b \) are:
\[ [b] = MT^{-2}L^{-1} \]

### Step 8: Calculate the dimensions of \( \frac{a}{b} \)
Now, we can find the dimensions of \( \frac{a}{b} \):
\[ \frac{a}{b} = \frac{T^2}{MT^{-2}L^{-1}} = \frac{T^2 \cdot L}{M \cdot T^{-2}} = \frac{L \cdot T^4}{M} \]

### Final Result
The dimensions of \( \frac{a}{b} \) are:
\[ \frac{a}{b} = \frac{L \cdot T^4}{M} \]