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Chapter 2: Problem 15

Business Bargains manufactures office supplies. It is considering selling sticky-notes in the shape of the state in which they will be sold. The expense and revenue functions are \(E=-250 p+50,000\) and \(R=-225 p^{2}+7,200 p .\) a. Determine the profit function. b. Determine the price, to the nearest cent, that yields the maximum profit. c. Determine the maximum profit, to the nearest cent.

Short Answer

Expert verified
The profit function is \(P = -225p^{2} + 7,450p - 50,000\), and the price that yields the maximum profit is \(\$16.56\). The maximum profit that can be obtained is \(\$60,994.80\) when rounded to the nearest cent.

Step by step solution

01

Finding the Profit Function

The profit function, \(P\), is found by subtracting the expense function, \(E\), from the revenue function, \(R\). This gives us: \(P = R - E = (-225 p^{2}+7,200 p ) - (-250 p+50,000 )\).
02

Simplifying the Profit Function

By simplifying the function obtained, we get: \(P = -225p^{2} + 7,200p + 250p - 50,000 = -225p^{2} + 7,450p - 50,000\). This is the profit function.
03

Finding the Derivative of the Profit function

To find the price that yields the maximum profit, first, the derivative of the profit function, \(P'\), needs to be calculated. The derivative of \(P\) with respect to \(p\) becomes: \(P' = -450p + 7,450\).
04

Equating the Derivative to Zero

Then, equate the derivative of the profit function to zero and solve for \(p\): \(-450p + 7,450 = 0 \)\(\Rightarrow p = 16.56\) after rounding to the nearest cent. This is the price that yields maximum profit.
05

Calculating the Maximum Profit

The maximum profit is obtained by substituting the price back into the profit function, \(P = -225(16.56)^{2} + 7,450(16.56) - 50,000 \) \(\Rightarrow P = 60,994.80\) after rounding to the nearest cent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Revenue Function
In business, the revenue function, denoted as \( R \), is crucial for determining how much money a company earns from selling a particular product. It generally depends on the price \( p \) and the quantity sold. In this exercise, the revenue function is given by \( R = -225 p^2 + 7,200 p \). What this tells us is a bit more complex than it appears at first glance. The quadratic nature of the function \( (-225 p^2) \) indicates that revenue is not linear; instead, it can show declining returns after reaching a peak at a certain price.
  • \( -225 p^2 \): This term illustrates the diminishing marginal returns as price increases. Higher prices could reduce sale volume hence lower revenue.
  • \( 7,200 p \): This term captures the increase in revenue with each unit increase in price up to a certain point.
Understanding how the revenue will increase or decrease with changes in price is pivotal for setting effective selling prices.
Delving into the Expense Function
The expense function \( E \), given by \( E = -250 p + 50,000 \), tells us about the company's cost structure. This function suggests that some expenses decrease with higher prices, highlighted by the \( -250 p \), and there are fixed costs represented by \( 50,000 \).
  • \( -250 p \): Represents variable costs that are inversely related to price. It suggests that higher prices could reduce variable expenses.
  • \( 50,000 \): This represents the fixed costs, which don't change with the level of production or the price set.
It's essential to evaluate both parts of the function. This helps in understanding how different pricing strategies will impact overall business expenses.
The Process of Derivative Calculation
To find the optimal price point where profit is maximized, we use calculus, specifically derivative calculation. The profit function \( P \), derived from \( P = R - E \), simplifies to \( P = -225p^2 + 7,450p - 50,000 \).
We find where the slope of this profit function equals zero, that is, where the rate of change in profit becomes zero, signaling potential maximum profit. The derivative, \( P' = -450p + 7,450 \), is calculated by differentiating the profit function.
  • \( -450p \): Represents how quickly profit changes with any infinitesimal change in price \( p \).
  • \( 7,450 \): Constant part of the slope showing revenue changes linearly at this point.
By setting \( P' = 0 \), we solve for \( p \), allowing us to find the exact price that maximizes profit.
Optimizing Price for Maximum Profit
Optimizing price for maximum profit is a critical step for any business. By calculating the derivative and setting it to zero, we find the value \( p = 16.56 \), which maximizes profit for Business Bargains. Substituting \( p = 16.56 \) back into the profit function provides us with the actual maximum obtainable profit.
  • The price \( p = 16.56 \) here is the **optimal price point**.
  • Substitute \( p \) in \( P = -225(16.56)^2 + 7,450(16.56) - 50,000 \) to find \( P = 60,994.80 \).
Businesses use this method not only to find how much should they charge for maximizing gains but also to understand their pricing strategies' impact on profitability.

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Most popular questions from this chapter

Orange-U-Happy is an orange-scented cleaning product that is manu- factured in disposable cloth pads. Fach box of 100 pads costs \(\$ 5\) to manufacture. The fixed costs for Orange-U-Happy are \(\$ 40,000\) . The research development group of the company has determined the demand function to be \(q=-5000\) , where \(p\) is the price for each box. a. Write the expense equation in terms of the demand, \(q\) . b. Express the expense function in terms of the price, \(p .\) c. Determine a viewing window on a graphing calculator for the expense function. Justify your answer. a. Draw and label the graph of the expense function. e. Write the revenue function in terms of the price. f. Graph the revenue function in a suitable viewing window. What price will yield the maximum revenue? What is the revenue at that price? Round answers to the nearest cent. g. Graph the revenue and expense functions on the same coordinate plane. Identify the points of intersection using a graphing calculator, and name the breakeven points. Round to the nearest cent. Identify the price at the breakeven points.

Flip Flops manufactures beach sandals. Their expense and revenue functions are \(E=-300 p+32,000\) and \(R=-275 p^{2}+6,500 p\) . a. Determine the profit function. b. Determine the price, to the nearest cent, that yields the maximum profi t. c. Determine the maximum profit, to the nearest cent.

Variable costs of producing widgets account for the cost of gas required to deliver the widgets to retailers. A widget producer finds the average cost of gas per widget. The expense equation was recently adjusted from \(E=4.55 q+69,000\) to \(E=4.98 q+69,000\) in response to the increase in gas prices. a. Find the increase in the average cost per widget. b. If the widgets are sold to retailers for \(\$ 8.00\) each, find the break even point prior to the adjustment in the expense function. c. After the gas increase, the company raised its wholesale cost from \(\$ 8\) to \(\$ 8.50\) . Find the breaker point after the adjustment in the expense function. Round to the nearest integer.

Use the following situation to answer Exercises 4鈥�20. A company produces a security device known as Toejack. Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child鈥檚 location at any time using an online system. The company has entered into an agreement with an Internet service provider, so the price of the chip will be low. Set up a demand function鈥攁 schedule of how many Toejacks would be demanded by the public at different prices. The horizontal axis represents price, and the vertical axis represents quantity. Does the demand function have a positive or negative slope? Explain.

Where-R-U produces global positioning systems (GPS) that can be used in a car. The expense equation is \(E=-5,000 p+\$ 8,300,000,\) and the revenue equation is \(R=-100 p^{2}+55,500 p\) . a. Graph the expense and revenue functions. Circle the breakeven points. b. Determine the prices at the breakeven points. Round to the nearest cent. c. Determine the revenue and expense amounts for each of the breakeven points. Round to the nearest cent.
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