Problem 18 - Applied Calculus

Question

A company's pricing schedule in Table 1.5 is designed to encourage large orders. (A gross is 12 dozen.) Find a formula for: (a) \(q\) as a linear function of \(p\) (b) \(p\) as a linear function of \(q\) \begin{array}{c|c|c|c|c} \hline q \text { (onder sixe, gross) } & 3 & 4 & 5 & 6 \\ \hline p \text { (price/dozen) } & 15 & 12 & 9 & 6 \\ \hline \end{array}

Step-by-Step Solution

Answer

(a) \(q = -\frac{1}{3}p + 8\); (b) \(p = -3q + 24\).

Step 1: Understanding Variables

In this exercise, we have a table correlating two variables: \(q\) (order size, in gross) and \(p\) (price per dozen). The relationship between \(q\) and \(p\) is given in the form of data pairs.

Step 2: Identify Linear Relationship Formulae

The general formula of a linear function is \(y = mx + b\). Here, we need to express \(q\) as a linear function of \(p\) and \(p\) as a linear function of \(q\). We start by identifying the slope \(m\) and the intercept \(b\) for each.

Step 3: Calculate the Slope for \(p\) as a Function of \(q\)

The slope \(m\) is calculated as \(m = \frac{\Delta p}{\Delta q}\). Using the data: when \(q\) increases from 3 to 6, \(p\) decreases from 15 to 6. So, \(m = \frac{6 - 15}{6 - 3} = -3\).

Step 4: Calculate Intercept for \(p(q)\)

Using the point \((q_1, p_1) = (3, 15)\) and the slope \(m = -3\), we find \(b\) using the formula: \(p = mq + b\). Plug in the numbers: \(15 = -3(3) + b\), giving \(b = 24\). Thus, \(p = -3q + 24\).

Step 5: Recalculate Slopes for \(q\) as a Function of \(p\)

To express \(q\) as a function of \(p\), consider the slope \(m = \frac{\Delta q}{\Delta p}\). From the previous data: \(m = \frac{6 - 3}{6 - 15} = -\frac{1}{3}\).

Step 6: Calculate Intercept for \(q(p)\)

Using the formula \(q = mp + b\) and point \((p_1, q_1) = (15, 3)\) with slope \(m = -\frac{1}{3}\), substitute to find \(b\): \(3 = -\frac{1}{3}(15) + b\), solving gives \(b = 8\). Thus, \(q = -\frac{1}{3}p + 8\).

Step 7: Provide Final Function Equations

The function of \(p\) as a function of \(q\) is \(p = -3q + 24\). Similarly, the function of \(q\) as a function of \(p\) is \(q = -\frac{1}{3}p + 8\).

Key Concepts

Pricing Schedule

A pricing schedule is a strategic table or listing designed by companies to communicate different price rates based on certain categories or scales.
In this specific problem, the pricing schedule shows the relation between the quantity ordered and the price per dozen. This schedule serves as a valuable tool for both the company and the customers. It encourages larger purchases by offering discounted prices for bulk orders, which is evident from the decrease in price per dozen as the order size increases.
This tactic is beneficial:

• Encouraging Larger Orders: The company incentivizes bulk purchasing by offering reduced prices. Customers pay less per dozen when they order more, thereby saving money.
• Boosting Sales Volume: With such a schedule, companies are likely to see an increase in sales volume as customers place larger orders to take advantage of the discounts.
• Simplifying Cost Planning: Predetermined pricing structures help customers plan their budgets more effectively.

In mathematics, such schedules can often be represented or analyzed using linear functions, enabling clearer understanding and easier calculations of the underlying relationships.

Slope-Intercept Form

Understanding the slope-intercept form can reveal the underlying patterns in a pricing schedule through a mathematical lens.
The slope-intercept form of a linear equation is written as \(y = mx + b\), where:

• \(m\) is the slope of the line, indicating the rate of change between the two variables.
• \(b\) is the y-intercept, representing the value of \(y\) when \(x\) is zero.

Using this form in our exercise, we derive two linear equations:

Price as a Function of Quantity (\(p(q)\)):
Here, the slope \(m\) is \(-3\). This negative value signifies that as the quantity orders (q) increases, the price per dozen (p) decreases. The y-intercept \(b\) is \(24\), which is the starting price per dozen when no packages are ordered. Thus, the function \(p(q) = -3q + 24\) allows us to calculate the price for different quantities with ease.

Quantity as a Function of Price (\(q(p)\)):
For the inverse relationship, the slope is \(-\frac{1}{3}\), and the intercept is \(8\). This setup helps find out how many dozens a customer would need to purchase to meet specific pricing thresholds. The function \(q(p) = -\frac{1}{3}p + 8\) shows the relationship in reverse, helping identify potential order sizes for given prices.

Variables Relationship

The relationship between variables in a linear equation reflects how changes in one variable impact the other within the given context. In the exercise, two main variables, \(q\) and \(p\), are interacting:

• Order Size (\(q\)): This represents the independent variable. Changes in \(q\) affect the price \(p\) directly in this pricing schedule.
• Price per Dozen (\(p\)): This is the dependent variable, influenced by the changes in the order quantity \(q\).

When dealing with linear functions, understanding this relationship helps in making accurate predictions and understanding the scalability of pricing models.
Let's explore how this relationship works:

• As \(q\) increases, \(p\) decreases due to the negative slope. This represents a typical discount strategy that rewards larger orders.
• The slope steepness in \(p(q)\) reflects how significantly the price drops with each added dozen.

This relational understanding allows for:

• Predictive Ordering: Determining how price adjustments can impact order size.
• Business Strategy: Companies analyzing how their pricing tactics can steer buying patterns.

Recognizing these changes can help businesses and customers adapt smarter strategies to optimize economic advantages.