At a price of $10 the local pizza sells 800 pizzas. At a price of $8 the pizza place sells 1000 pizzas. When the quantity equals 100 pizzas, total costs equaled $1300. When quantity equaled 200 pizzas, total costs equaled $ 2200. When quantity equaled 300 total costs equal $3300.

Question

At a price of $10 the local pizza sells 800 pizzas. At a price of $8 the pizza place sells 1000 pizzas. When the quantity equals 100 pizzas, total costs equaled $1300. When quantity equaled 200 pizzas, total costs equaled $ 2200. When quantity equaled 300 total costs equal $3300.

Q.
1.Develop a total revenue function
2.Develop a total cost function
3.Determine the quantity that maximises profit
4.Determine the price that maximises profit
5.Determine breakeven points

Answer 1

To answer these questions, we will need to use the concepts of total revenue, total cost, and profit. Let's work through each question step by step:

1. To develop a total revenue function, we need to multiply the quantity of pizzas sold by the price at which they are sold. From the given information, we know that at a price of $10, 800 pizzas were sold, and at a price of $8, 1000 pizzas were sold. So, the total revenue (R) function can be written as:

R = price * quantity

For the given scenario, the total revenue function can be represented as:

R = 10Q if Q <= 800
R = 8Q if 800 < Q <= 1000

2. To develop a total cost function, we need to find a relationship between the quantity of pizzas and their total costs. From the given information, we know that when the quantity is 100, the total cost is $1300, when the quantity is 200, the total cost is $2200, and when the quantity is 300, the total cost is $3300. We can use this information to find the equation of the total cost (C) function.

Let's denote the fixed cost as F and the variable cost per pizza as VC. The total cost function can be represented as:

C = F + VC * quantity

Now, we can substitute the given values to find the equation of the total cost function. Using the points (100, 1300) and (200, 2200), we can solve the system of equations to find the values of F and VC.

Substituting the first point (100, 1300):
1300 = F + 100 * VC ----> Equation 1

Substituting the second point (200, 2200):
2200 = F + 200 * VC ----> Equation 2

Solving this system of equations will give us the values of F and VC, which will allow us to develop the total cost function.

3. To determine the quantity that maximizes profit, we need to compare the total revenue and total cost functions. The profit (P) can be calculated as the difference between total revenue and total cost:

P = R - C

To find the quantity that maximizes profit, we can take the derivative of the profit function with respect to quantity (dP/dQ) and set it equal to zero. Solving this equation will give us the quantity that maximizes profit.

4. To determine the price that maximizes profit, we can follow a similar approach as in the previous question. We can take the derivative of the profit function with respect to price (dP/dP), set it equal to zero, and solve for the price that maximizes profit.

5. To determine the breakeven points, we need to find the quantity at which the total revenue equals the total cost. In other words, we want to find the quantity at which P = 0 (break-even). We can set the profit function equal to zero (P = R - C = 0), substitute the known revenue and cost functions, and solve for the quantity. The resulting quantity will give us the breakeven points.