Final answer:
The total profit of producing 2 license plates is -35 dollars. The marginal profit of producing the 9th license plate is 9p - 62 dollars. The average profit of producing 3 license plates is -12.67 dollars per license plate. The marginal average profit of producing the 7th license plate is p - 8.57 dollars.
Step-by-step explanation:
To calculate the total profit of producing 2 license plates, we need to use the total cost function and the price-demand equation. The total cost function C(x) = 53 + 5x gives us the cost of producing x license plates. Plugging in x = 2, we get C(2) = 53 + 5(2) = 63 dollars. Next, we can use the price-demand equation 4x + p = 22 to solve for the price p. Plugging in x = 2, we get 4(2) + p = 22. Solving for p, we find p = 14. Therefore, the total profit of producing 2 license plates is the difference between the revenue and the cost, which is (p * x) - C(x) = (14 * 2) - 63 = 28 - 63 = -35 dollars.
To calculate the marginal profit of producing the 9th license plate, we need to find the change in profit when the quantity increases from 8 to 9. The total profit function is given by (p * x) - C(x). Plugging in x = 9, we get (p * 9) - C(9) = (4(9) + p * 9) - (53 + 5 * 9). Simplifying this expression, we get 36 + 9p - 53 - 45 = 9p - 62. Therefore, the marginal profit of producing the 9th license plate is 9p - 62 dollars.
To calculate the average profit of producing 3 license plates, we first need to find the total profit of producing 3 license plates. We can use the total cost function and the price-demand equation as before to find the total profit. Plugging in x = 3, we get C(3) = 53 + 5(3) = 68 dollars. Plugging in x = 3 into the price-demand equation, we get 4(3) + p = 22. Solving for p, we find p = 10. Therefore, the total profit of producing 3 license plates is (p * x) - C(x) = (10 * 3) - 68 = 30 - 68 = -38 dollars. Finally, the average profit is the total profit divided by the quantity, which is -38 / 3 = -12.67 dollars per license plate.
To calculate the marginal average profit of producing the 7th license plate, we need to find the change in average profit when the quantity increases from 6 to 7. The average profit function is given by (p * x) - C(x) / x. Plugging in x = 7, we get (p * 7) - C(7) / 7 = (4(7) + p * 7) - (53 + 5 * 7) / 7. Simplifying this expression, we get (28 + 7p - 53 - 35) / 7 = (7p - 60) / 7 = p - 8.57. Therefore, the marginal average profit of producing the 7th license plate is p - 8.57 dollars.