Fran owns a candle-making business. Each product sells for $8, and Fran sells to 4,000 people each month. She wants to make a profit and wants to determine whether she should raise the price in increments of a dollar. For each dollar she raises the price, she will lose 80 customers. Part A: Write an equation to represent the maximum profit in standard form. Show your work.

Part B: Write an equation to represent the maximum profit in vertex form. Show your work.

Part C: How many dollars can Fran raise the price of her items to maximize her profit?

Part A:

The equation for the maximum profit in standard form is y = 3360·x - 80·x²

Part B:

The equation for the maximum profit in vertex form is y = -80·(x - 21)² + 35280

Part C: The amount of dollars Fran can raise the price of her items to maximize her profit is given by value of x at the vertex, which is 21 dollars

Step-by-step explanation:

Part A:

The current price for each product = $8

The number of customers for the product = 4,000

The number of customers lost per dollar rice in the product price = 80

Total revenue = f(x) = (4000 - 80·x) × (8 + x) = -80×(x² - 42·x -400)

Maximum Profit = Total revenue - Cost = -80×(x² - 42·x -400) - 4000×8

Profit, y = 3360·x - 80·x²

The equation for the maximum profit in standard form is y = 3360·x - 80·x²

Part B:

The maximum profit is given by the peak of the graph of the function for the profit, which is obtained by differentiating the function and equating to zero as follows;

dy/dx = d(3360·x - 80·x²)/dx = 3360 - 160·x = 0

3360 = 160·x

x = 3360/160 = 21

x = 21

f(21) = y₂₁ = 3360×21 - 80×21² = 35,280

The vertex = (21, 35,280)

Therefore, the equation for the maximum profit in vertex form is -80·(x - 21)² + 35280

Part C: The amount of dollars Fran can raise the price of her items to maximize her profit is given by value of x at the vertex, which is x = 21 dollars.