Problem 66 - Algebra and Trigonometry

Question

Internet Purchases An Internet bookstore charges \(\$ 15\) shipping for orders under \(\$ 100\) , but provides free shipping for orders of \(\$ 100\) or more. The cost \(C\) of an order is a function of the total price \(x\) of the books purchased, given by $$ C(x)=\left\\{\begin{array}{ll}{x+15} & {\text { if } x<100} \\ {x} & {\text { if } x \geq 100}\end{array}\right. $$ (a) Find \(C(75), C(90), C(100),\) and \(C(105)\) (b) What do your answers in part (a) represent?

Step-by-Step Solution

Answer

For orders below \$100, add \$15 shipping; for \$100 or more, no shipping cost.

Step 1: Understand the Problem

To find the cost of an order, we need to apply the function \(C(x)\). It has two parts: if the total price of books \(x\) is less than \\(100, then the shipping fee of \\)15 is added; if it is \$100 or more, shipping is free.

Step 2: Calculate for C(75)

Since \(75 < 100\), we use the first part of the function: \(C(x) = x + 15\). Substitute \(x = 75\) to get \(C(75) = 75 + 15 = 90\).

Step 3: Calculate for C(90)

Since \(90 < 100\), we again use \(C(x) = x + 15\). Substitute \(x = 90\) to get \(C(90) = 90 + 15 = 105\).

Step 4: Calculate for C(100)

Since \(100 \geq 100\), we use the second part of the function: \(C(x) = x\). Thus, \(C(100) = 100\).

Step 5: Calculate for C(105)

Since \(105 \geq 100\), we use \(C(x) = x\). Therefore, \(C(105) = 105\).

Step 6: Interpret the Results

The values \(C(75) = 90\), \(C(90) = 105\), \(C(100) = 100\), and \(C(105) = 105\) represent the total cost of orders, including shipping for orders under \\(100 and free shipping for orders of \\)100 or more.

Key Concepts

Cost Function

A cost function is a mathematical expression or model that helps to determine the total cost associated with a specific amount of purchases or activities. In the context of the Internet bookstore problem, the cost function is defined as a piecewise function, which means it has different expressions based on the value of the variable. Here, the cost function is:- For purchases under \(100, there's an additional \)15 charge for shipping, i.e., \( C(x) = x + 15 \)- For purchases of $100 or more, shipping is free, hence \( C(x) = x \)This setup ensures that customers clearly understand how their total costs vary depending on their total book purchase. The cost function effectively accounts for whether shipping fees apply, thus highlighting how physical shipping costs impact the overall pricing structure in e-commerce.

Step-by-Step Solution

Let’s break down the steps to compute the cost for different order values using the piecewise cost function.1. **Understand the Function**: The task is to apply the appropriate part of the piecewise function based on the value of the order: - Use \( C(x) = x + 15 \) if \( x < 100 \) - Use \( C(x) = x \) if \( x \geq 100 \)2. **Calculate Specific Costs**: - For \( C(75) \): Since 75 is less than 100, apply the first formula. Calculate: \( C(75) = 75 + 15 = 90 \). - For \( C(90) \): Again, 90 is less than 100, so use \( C(x) = x + 15 \). Calculate: \( C(90) = 90 + 15 = 105 \). - For \( C(100) \): Here, 100 is equal to 100, thus use the second formula: \( C(x) = x \). Calculate: \( C(100) = 100 \). - For \( C(105) \): Since 105 is greater than 100, apply \( C(x) = x \) again. Calculate: \( C(105) = 105 \).3. **Results Interpretation**: The results \( C(75) = 90, C(90) = 105, C(100) = 100, C(105) = 105 \) each represent the total costs of orders, factoring in shipping costs for certain order values. This explanation ensures understanding of how different order values affect total cost based on the presence or absence of a shipping fee.

Inequality

To determine which part of the piecewise function to use, you need to evaluate inequalities. Inequalities are expressions used to compare values, indicating whether one number is greater than, less than, or equal to another.- For orders where the price \( x \) of books is **less than 100** (\( x < 100 \)), the inequality dictates using the part of the function \( C(x) = x + 15 \). - Example: \( C(75) \) and \( C(90) \) fall into this category, leading to the addition of a shipping cost.- When the order total is **100 or more** (\( x \geq 100 \)), it satisfies the condition to utilize \( C(x) = x \), reflecting the eligibility for free shipping. - Example: \( C(100) \) and \( C(105) \) fulfill this inequality, so the cost function does not include additional fees.Understanding how to set up and evaluate these inequalities helps in choosing the correct part of the function to apply, ensuring accurate computation of total costs based on the presence of conditional shipping.