Final answer:
To calculate the profit of the monopolist, subtract the total cost from the total revenue. Based on the given information, the fixed cost is $12 and the variable cost function is VC = 2Q + 0.2Q². By substituting the quantities into the variable cost function and calculating the total revenue and total cost, the profit can be determined. None of the given options match the calculated profits.
Step-by-step explanation:
To calculate the profit of the monopolist, we need to subtract the total cost from the total revenue. Based on the given information, the fixed cost is $12. The variable cost function is VC = 2Q + 0.2Q². To find the total cost, we need to calculate the sum of the fixed cost and the variable cost at the given quantity. The total revenue is the price multiplied by the quantity. By subtracting the total cost from the total revenue, we can find the profit.
Let's substitute the given quantity into the variable cost function:
VC = 2Q + 0.2Q²
Now, calculate the total cost of each quantity:
- Q = 1: TC = $12 + (2 * 1) + (0.2 * 1²) = $14.2
- Q = 2: TC = $12 + (2 * 2) + (0.2 * 2²) = $18.8
- Q = 3: TC = $12 + (2 * 3) + (0.2 * 3²) = $26.4
- Q = 4: TC = $12 + (2 * 4) + (0.2 * 4²) = $37.2
- Q = 5: TC = $12 + (2 * 5) + (0.2 * 5²) = $51.2
Now, calculate the total revenue for each quantity (TR = price * quantity):
- Q = 1: TR = $800 * 1 = $800
- Q = 2: TR = $800 * 2 = $1600
- Q = 3: TR = $800 * 3 = $2400
- Q = 4: TR = $800 * 4 = $3200
- Q = 5: TR = $800 * 5 = $4000
Finally, calculate the profit by subtracting the total cost from the total revenue:
- Q = 1: Profit = $800 - $14.2 = $785.8
- Q = 2: Profit = $1600 - $18.8 = $1581.2
- Q = 3: Profit = $2400 - $26.4 = $2373.6
- Q = 4: Profit = $3200 - $37.2 = $3162.8
- Q = 5: Profit = $4000 - $51.2 = $3948.8
Based on the calculations, the profit of the monopolist varies based on the quantity sold. None of the given options (a. $350, b. $420, c. $668, d. $639) match the calculated profits. Therefore, none of the given options is correct.