The marginal cost of the production of the commodity is `30+2x`, it is known that fixed costs are Rs 200, find
(i) The total cost.
(ii) The cost of increasing output from 100 to 200 units.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Marginal Cost Function
The marginal cost (MC) of production is given as:
\[ MC = 30 + 2x \]
where \( x \) is the quantity of output produced.

### Step 2: Find the Total Cost Function
The total cost (TC) can be found by integrating the marginal cost function. Since marginal cost is the derivative of total cost with respect to quantity, we have:
\[ \frac{d(TC)}{dx} = MC \]
Thus, we can write:
\[ d(TC) = (30 + 2x)dx \]

Now, we integrate both sides:
\[ TC = \int (30 + 2x) \, dx \]
Calculating the integral:
\[ TC = 30x + x^2 + C \]
where \( C \) is the constant of integration.

### Step 3: Include Fixed Costs
We know that fixed costs are Rs 200. Therefore, we can express the total cost as:
\[ TC = 200 + 30x + x^2 \]

### Step 4: Calculate the Total Cost for Specific Outputs
(i) The total cost function is:
\[ TC = 200 + 30x + x^2 \]

(ii) To find the cost of increasing output from 100 to 200 units, we need to calculate the total cost at these two levels of output.

1. **Total Cost at 200 units:**
\[ TC(200) = 200 + 30(200) + (200)^2 \]
\[ TC(200) = 200 + 6000 + 40000 = 46200 \]

2. **Total Cost at 100 units:**
\[ TC(100) = 200 + 30(100) + (100)^2 \]
\[ TC(100) = 200 + 3000 + 10000 = 13200 \]

### Step 5: Calculate the Cost of Increasing Output
The cost of increasing output from 100 to 200 units is:
\[ \text{Cost Increase} = TC(200) - TC(100) \]
\[ \text{Cost Increase} = 46200 - 13200 = 33000 \]

### Final Answers
(i) The total cost function is:
\[ TC = 200 + 30x + x^2 \]

(ii) The cost of increasing output from 100 to 200 units is Rs 33000.

---