The total cost function of a firm is `C= (5)/(3) x^(3)-10x^(2) + 32x + 15`, where C is the total cost and x is the output. A tax at the rate of Rs2 per unit is imposed by the Govermment and the producer adds it to its cost. Demand function is given by `p= 4534-10x`, where p is price per unit of the output. Find the profit function

Answer

To find the profit function for the given total cost and demand functions, we will follow these steps:

### Step 1: Write down the total cost function
The total cost function is given as:
\[ C(x) = \frac{5}{3} x^3 - 10x^2 + 32x + 15 \]

### Step 2: Adjust the total cost for the tax
Since a tax of Rs 2 per unit is added to the cost, we need to adjust the total cost function. The new total cost function becomes:
\[ C(x) = \frac{5}{3} x^3 - 10x^2 + (32 + 2)x + 15 \]
This simplifies to:
\[ C(x) = \frac{5}{3} x^3 - 10x^2 + 34x + 15 \]

### Step 3: Write down the demand function
The demand function is given as:
\[ p = 4534 - 10x \]

### Step 4: Calculate the revenue function
Revenue (R) is calculated as the product of price (p) and quantity (x):
\[ R(x) = p \cdot x = (4534 - 10x) \cdot x \]
This expands to:
\[ R(x) = 4534x - 10x^2 \]

### Step 5: Write the profit function
The profit function (P) is defined as the difference between revenue and total cost:
\[ P(x) = R(x) - C(x) \]
Substituting the expressions we found:
\[ P(x) = (4534x - 10x^2) - \left(\frac{5}{3} x^3 - 10x^2 + 34x + 15\right) \]

### Step 6: Simplify the profit function
Now, we simplify the profit function:
\[ P(x) = 4534x - 10x^2 - \frac{5}{3} x^3 + 10x^2 - 34x - 15 \]
The \( -10x^2 \) and \( +10x^2 \) cancel each other out:
\[ P(x) = 4534x - 34x - \frac{5}{3} x^3 - 15 \]
This simplifies to:
\[ P(x) = (4534 - 34)x - \frac{5}{3} x^3 - 15 \]
Calculating \( 4534 - 34 \):
\[ P(x) = 4500x - \frac{5}{3} x^3 - 15 \]

### Final Profit Function
Thus, the profit function is:
\[ P(x) = 4500x - \frac{5}{3} x^3 - 15 \]

---