The marginal cost MC of a product is given to be a constant multiple of number of units (x) produced. Find the total cost function if the fixed cost is Rs. 1000 and the cost of producing 30 units is Rs. 2800.

Answer

To solve the given problem, we need to find the total cost function based on the information provided about marginal cost, fixed costs, and the cost of producing a certain number of units. Let's break it down step by step.

### Step 1: Understand the Marginal Cost
The marginal cost (MC) is given as a constant multiple of the number of units produced (x). We can express this as:
\[ MC = kx \]
where \( k \) is a constant.

### Step 2: Relate Marginal Cost to Total Cost
Marginal cost is the derivative of the total cost function (C) with respect to the number of units (x):
\[ MC = \frac{dC}{dx} \]
Thus, we have:
\[ \frac{dC}{dx} = kx \]

### Step 3: Integrate to Find Total Cost Function
To find the total cost function (C), we integrate the marginal cost:
\[ dC = kx \, dx \]
Integrating both sides gives:
\[ C = \frac{kx^2}{2} + C_1 \]
where \( C_1 \) is the constant of integration.

### Step 4: Use Fixed Cost Information
We know that the fixed cost is Rs. 1000. This means that when no units are produced (x = 0), the total cost (C) is equal to the fixed cost:
\[ C(0) = 1000 \]
Substituting \( x = 0 \) into the total cost function:
\[ 1000 = \frac{k(0)^2}{2} + C_1 \]
This simplifies to:
\[ C_1 = 1000 \]

### Step 5: Update the Total Cost Function
Now we can update our total cost function:
\[ C = \frac{kx^2}{2} + 1000 \]

### Step 6: Use Additional Information to Find k
We are given that the cost of producing 30 units is Rs. 2800. Therefore:
\[ C(30) = 2800 \]
Substituting \( x = 30 \) into the total cost function:
\[ 2800 = \frac{k(30)^2}{2} + 1000 \]
This simplifies to:
\[ 2800 = \frac{k(900)}{2} + 1000 \]
\[ 2800 = 450k + 1000 \]

### Step 7: Solve for k
Now we can isolate k:
\[ 2800 - 1000 = 450k \]
\[ 1800 = 450k \]
\[ k = \frac{1800}{450} = 4 \]

### Step 8: Finalize the Total Cost Function
Now that we have the value of k, we can substitute it back into the total cost function:
\[ C = \frac{4x^2}{2} + 1000 \]
This simplifies to:
\[ C = 2x^2 + 1000 \]

### Final Answer
Thus, the total cost function is:
\[ C(x) = 2x^2 + 1000 \]

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