Suppose the cost to produce some commodity is linear function of output, find cost as a function of output, if cost are रु4000 for 250 units and रु5000 for 350 units. Hence find marginal cost.

Answer

To solve the problem, we need to find the cost function \( C(x) \) as a linear function of output \( x \), given two points: when \( x = 250 \), \( C = 4000 \) and when \( x = 350 \), \( C = 5000 \).

### Step-by-Step Solution:

1. **Assume the Cost Function**:
Since the cost is a linear function of output, we can express it as:
\[ C(x) = ax + b \]
where \( a \) is the slope (marginal cost), and \( b \) is the fixed cost.

2. **Set Up the Equations**:
From the given information, we can set up two equations based on the cost at the given outputs:
- For \( x = 250 \):
\[ C(250) = 4000 \implies 250a + b = 4000 \quad \text{(Equation 1)} \]
- For \( x = 350 \):
\[ C(350) = 5000 \implies 350a + b = 5000 \quad \text{(Equation 2)} \]

3. **Subtract the Equations**:
To eliminate \( b \), we can subtract Equation 1 from Equation 2:
\[ (350a + b) - (250a + b) = 5000 - 4000 \]
This simplifies to:
\[ 100a = 1000 \]
Therefore, solving for \( a \):
\[ a = \frac{1000}{100} = 10 \]

4. **Substitute \( a \) Back to Find \( b \)**:
Now that we have \( a \), we can substitute it back into Equation 1 to find \( b \):
\[ 250(10) + b = 4000 \]
\[ 2500 + b = 4000 \]
\[ b = 4000 - 2500 = 1500 \]

5. **Write the Cost Function**:
Now we can write the cost function:
\[ C(x) = 10x + 1500 \]

6. **Find the Marginal Cost**:
The marginal cost is the derivative of the cost function with respect to \( x \):
\[ C'(x) = \frac{d}{dx}(10x + 1500) = 10 \]

### Final Results:
- The cost function is:
\[ C(x) = 10x + 1500 \]
- The marginal cost is:
\[ \text{Marginal Cost} = 10 \]