Question

Orange-U-Happy is an orange-scented cleaning product that is manufactured in disposable cloth pads. Each box of 100 pads costs $5 to manufacture. The fixed costs for range-U-Happy are $40,000. The research development group of the company has determined the demand function to be q = –500p + 20,000, where p is the price for each box. Part A Create expense and revenue functions to model the business in terms of the price, p. Part B What is the breakeven point for Orange-U-Happy’s disposable cloth pads? Explain how you determined your answer. Part C Create the profit function for Orange-U-Happy in terms of the price, p. Part D Determine the maximum profit and the price at which the maximum profit occurs for the disposable cloth pads. Explain how you determined your answer.

Answer 1

To model Orange-U-Happy's business in mathematical terms, we create expense and revenue functions based on the price per box, find the breakeven point by equating these functions, and derive a profit function to determine the maximum profit and the price at which it occurs.

Step-by-step explanation:

Part A: Expense and Revenue Functions

To create the expense and revenue functions for Orange-U-Happy in terms of the price p, we define:

Expense function: Cost = Fixed costs + (Variable cost per unit × Quantity)

Revenue function: Revenue = Price per box × Quantity sold

The fixed costs are $40,000, and the variable cost per unit is $5. The demand function is given as q = –500p + 20,000. We can now express both functions in terms of p as follows:

Expense function: Cost(p) = $40,000 + ($5 × q)

Revenue function: Revenue(p) = p ×

Since q is defined by the demand function, we substitute –500p + 20,000 in place of q to get the functions in terms of p only.

Expense function: Cost(p) = $40,000 + ($5 × (–500p + 20,000))

Revenue function: Revenue(p) = p × (–500p + 20,000)

Part B: Breakeven Point

To find the breakeven point, we need to set the expense function equal to the revenue function and solve for p. This gives us the price point at which the company neither makes a profit nor incurs a loss. After equating and simplifying the functions, we can find the value of p that satisfies this condition.

Part C: Profit Function

The profit function is found by subtracting the cost from the revenue, which we can express as:

Profit(p) = Revenue(p) – Cost(p

Part D: Maximum Profit

To determine the maximum profit and the price at which it occurs, we need to analyze the profit function. We can either factor the quadratic profit function and find its vertex, or use calculus to find the derivative of the profit function, set it to zero, and solve for p. The p that maximizes the profit is the price to charge for each box, and the corresponding profit value is the maximum profit.