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Chapter 2: Problem 20

An enterprise rents out paddleboats for all-day use on a lake. The owner knows that he can rent out all 27 of his paddleboats if he charges \(\$ 1\) for each rental. He also knows that he can rent out only 26 if he charges \(\$ 2\) for each rental and that, in general, there will be 1 less paddleboat rental for each extra dollar he charges per rental. a. What would the owner's total revenue be if he charged \(\$ 3\) for each paddleboat rental? b. Use a formula to express the number of rentals as a function of the amount charged for each rental. c. Use a formula to express the total revenue as a function of the amount charged for each rental. d. How much should the owner charge to get the largest total revenue?

Short Answer

Expert verified
a. $75; b. f(p) = 28 - p; c. R(p) = 28p - p^2; d. Charge $14 for maximum revenue.

Step by step solution

01

Understand the Revenue Relationship

Each additional dollar charged results in 1 fewer rental. Initially, all 27 boats are rented for $1 each. For every dollar increase in price, the number of rentals decreases by 1.
02

Revenue Calculation for $3 per Rental

If the owner charges \(3, he rents out \(27 - 2 = 25\) paddleboats (since the price increased by \)2 from \(1). Therefore, the total revenue at \)3 per rental is computed as: \(3 \times 25 = 75\).
03

Create a Rentals Function

Let \(p\) represent the price charged per rental. The number of rentals as a function of \(p\) is given by \(f(p) = 27 - (p - 1)\). Simplifying, \(f(p) = 28 - p\).
04

Create a Total Revenue Function

Total revenue \(R(p)\) can be expressed as a function of the price \(p\) by multiplying the price per rental by the number of rentals. Substituting \(f(p)\), we have: \[ R(p) = p \times (28 - p) = 28p - p^2 \].
05

Optimize Revenue Function

To find the price that maximizes revenue, we need to find the vertex of the quadratic function \(R(p) = -p^2 + 28p\). The vertex formula \(p = -\frac{b}{2a}\) for \(ax^2 + bx + c\) is used. Here, \(a = -1,\ b = 28\), so \(p = -\frac{28}{2 \times -1} = 14\).

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Key Concepts

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

Quadratic Function
A quadratic function is a type of algebraic expression that can be written in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This type of function creates a parabolic shape when graphed on a coordinate plane.

In the context of the paddleboat rental problem, our function for revenue, \( R(p) = -p^2 + 28p \), is a classic example of a quadratic function. The coefficient \( a = -1 \) creates a downward opening parabola, indicating that there is a maximum point or vertex, rather than a minimum.

Quadratic functions are often used in applications including business, physics, and engineering because they can model a wide variety of real-world situations. They are particularly useful for problems involving maximum or minimum values, which are often found at the vertex of the parabola created by the function.
Function Optimization
Function optimization involves finding the maximum or minimum value of a function. In this exercise, we focus on maximizing the total revenue of renting out paddleboats.

To achieve this, we use the vertex of the quadratic function \( R(p) = -p^2 + 28p \). The vertex represents the peak point of the parabola, giving us the price \( p \) that results in the maximum revenue.

The vertex formula is \( p = -\frac{b}{2a} \), applicable to any quadratic form \( ax^2 + bx + c \). Here, by substituting \( a = -1 \) and \( b = 28 \), we find \( p = 14 \). This tells us that charging \$14 per rental maximizes revenue.

This process of finding extreme values is critical in business and economics, where optimization can greatly affect profitability and resource allocation.
Business Mathematics
Business mathematics encompasses various mathematical techniques and concepts applied to solve real business issues. In our paddleboat rental problem, we use these concepts to assess and optimize financial decisions.

Here, revenue maximization is the goal. Understanding the functional relationship between price and rentals allows us to create 鈥� models and predict outcomes.
鈥� Customers' sensitivity to price changes.
鈥� The ramifications of price increases on total revenue.

By setting up a revenue model \( R(p) = 28p - p^2 \), business mathematics enables the prediction of how different pricing strategies impact the total income.

Applying these concepts, business owners can make informed decisions that balance customer demand and operational capacity, leading to optimal revenue generation and improved business strategies.

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Most popular questions from this chapter

A scientist observed that the speed \(S\) at which certain ants ran was a function of \(T\), the ambient temperature. \({ }^{17}\) He discovered the formula $$ S=0.2 T-2.7, $$ where \(S\) is measured in centimeters per second and \(T\) is in degrees Celsius. a. Using functional notation, express the speed of the ants when the ambient temperature is 30 degrees Celsius, and calculate that speed using the formula above. b. Solve for \(T\) in the formula above to obtain a formula expressing the ambient temperature \(T\) as a function of the speed \(S\) at which the ants run. c. If the ants are running at a speed of 3 centimeters per second, what is the ambient temperature?

The monthly profit \(P\) for a widget producer is a function of the number \(n\) of widgets sold. The formula is $$ P=-15+10 n-0.2 n^{2} . $$ Here \(P\) is measured in thousands of dollars, \(n\) is measured in thousands of widgets, and the formula is valid up to a level of 15 thousand widgets sold. a. Make a graph of \(P\) versus \(n\). b. Calculate \(P(1)\) and explain in practical terms what your answer means. c. Is the graph concave up or concave down? Explain in practical terms what this means. d. The break-even point is the sales level at which the profit is 0 . Find the break-even point for this widget producer.

For retailers who buy from a distributor or manufacturer and sell to the public, a major concern is the cost of maintaining unsold inventory. You must have appropriate stock to do business, but if you order too much at a time, your profits may be eaten up by storage costs. One of the simplest tools for analysis of inventory costs is the basic or der quantity model. It gives the yearly inventory expense \(E=E(c, N, Q, f)\) when the following inventory and restocking cost factors are taken into account: \- The carrying cost \(c\), which is the cost in dollars per year of keeping a single unsold item in your warehouse. \- The number \(N\) of this item that you expect to sell in 1 year. \- The number \(Q\) of items you order at a time. \- The fixed costs \(f\) in dollars of processing a restocking order to the manufacturer. (Note: This is not the cost of the order; the price of an item does not play a role here. Rather, \(f\) is the cost you would incur with any order of any size. It might include the cost of processing the paperwork, fixed costs you pay the manufacturer for each order, shipping charges that do not depend on the size of the order, the cost of counting your inventory, or the cost of cleaning and rearranging your warehouse in preparation for delivery.) The relationship is given by $$ E=\left(\frac{Q}{2}\right) c+\left(\frac{N}{Q}\right) f \text { dollars per year. } $$ A new-car dealer expects to sell 36 of a particular model car in the next year. It costs \(\$ 850\) per year to keep an unsold car on the lot. Fixed costs associated with preparing, processing, and receiving a single order from Detroit total \(\$ 230\) per order. a. Using the information provided, express the yearly inventory expense \(E=E(Q)\) as a function of \(Q\), the number of automobiles included in a single order. b. What is the yearly inventory expense if 3 cars at a time are ordered? c. How many cars at a time should be ordered to make yearly inventory expenses a minimum? d. Using the value of \(Q\) you found in part c, determine how many orders to Detroit will be placed this year. e. What is the average rate of increase in yearly inventory expense from the number you found in part \(\mathrm{c}\) to an order of 2 cars more?

This is a continuation of Exercise 7. In many situations, the number of possibilities is not affected by order. For example, if a group of 4 people is selected from a group of 20 to go on a trip, then the order of selection does not matter. In general, the number \(C\) of ways to select a group of \(k\) things from a group of \(n\) things is given by $$ C=\frac{n !}{k !(n-k) !} $$ if \(k\) is not greater than \(n\). a. How many different groups of 4 people could be selected from a group of 20 to go on a trip? b. How many groups of 16 could be selected from a group of 20 ? c. Your answers in parts a and b should have been the same. Explain why this is true. d. What group size chosen from among 20 people will result in the largest number of possibilities? How many possibilities are there for this group size?

The weekly cost of running a small firm is a function of the number of employees. Every week there is a fixed cost of \(\$ 2500\), and each employee costs the firm \(\$ 350\). For example, if there are 10 employees, then the weekly cost is \(2500+350 \times 10=6000\) dollars. a. What is the weekly cost if there are 3 employees? b. Find a formula for the weekly cost as a function of the number of employees. (You need to choose variable and function names. Be sure to state the units.) c. Make a graph of the weekly cost as a function of the number of employees. Include values of the variable up to 10 employees. d. For what number of employees will the weekly cost be \(\$ 4250\) ?
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