The dimension of impulse is `[M^(2b)L^(3a)T^(4c)]`.Find the value of `3a+2b-4c`

Explanation

To solve the problem, we need to find the value of \(3a + 2b - 4c\) given the dimension of impulse as \([M^{2b}L^{3a}T^{4c}]\).

### Step 1: Understand the definition of impulse
Impulse is defined as the product of force and time. Mathematically, it can be expressed as:
\[ \text{Impulse} = \text{Force} \times \text{Time} \]

### Step 2: Find the dimension of force
The dimension of force can be derived from Newton's second law, which states that force is mass times acceleration:
\[ \text{Force} = \text{Mass} \times \text{Acceleration} \]
The dimension of mass is \([M]\) and the dimension of acceleration is \([L T^{-2}]\). Therefore, the dimension of force is:
\[ \text{Dimension of Force} = [M][L T^{-2}] = [M^1 L^1 T^{-2}] \]

### Step 3: Find the dimension of impulse
Since impulse is force multiplied by time, we can write:
\[ \text{Impulse} = \text{Force} \times \text{Time} \]
The dimension of time is \([T]\). Thus, the dimension of impulse becomes:
\[ \text{Dimension of Impulse} = [M^1 L^1 T^{-2}] \times [T^1] = [M^1 L^1 T^{-1}] \]

### Step 4: Set up the equation with given dimensions
We are given that the dimension of impulse is \([M^{2b}L^{3a}T^{4c}]\). Therefore, we can equate the dimensions:
\[ [M^1 L^1 T^{-1}] = [M^{2b} L^{3a} T^{4c}] \]

### Step 5: Compare the powers of M, L, and T
From the equation above, we can compare the powers of \(M\), \(L\), and \(T\):
1. For \(M\):
\[ 2b = 1 \quad \Rightarrow \quad b = \frac{1}{2} \]
2. For \(L\):
\[ 3a = 1 \quad \Rightarrow \quad a = \frac{1}{3} \]
3. For \(T\):
\[ 4c = -1 \quad \Rightarrow \quad c = -\frac{1}{4} \]

### Step 6: Calculate \(3a + 2b - 4c\)
Now we can substitute the values of \(a\), \(b\), and \(c\) into the expression \(3a + 2b - 4c\):
\[ 3a + 2b - 4c = 3\left(\frac{1}{3}\right) + 2\left(\frac{1}{2}\right) - 4\left(-\frac{1}{4}\right) \]
Calculating each term:
- \(3a = 3 \times \frac{1}{3} = 1\)
- \(2b = 2 \times \frac{1}{2} = 1\)
- \(-4c = -4 \times -\frac{1}{4} = 1\)

Adding these together:
\[ 3a + 2b - 4c = 1 + 1 + 1 = 3 \]

### Final Answer
Thus, the value of \(3a + 2b - 4c\) is:
\[ \boxed{3} \]