Problem 67 - 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

(a) \( C(75) = 90 \), \( C(90) = 105 \), \( C(100) = 100 \), \( C(105) = 105 \). (b) The results show total costs with shipping for various book prices.

Step 1: Identify the Function

We have the cost function: \[ \begin{cases} x + 15 & \text{if } x < 100 \ x & \text{if } x \geq 100 \end{cases} \]This means that the cost of an order, \(C(x)\), involves adding a \\(15 shipping fee if the total price of books, \(x\), is less than \\)100. If \(x\) is \$100 or more, there is no shipping fee.

Step 2: Find C(75)

Since \( 75 < 100 \), we use the first part of the function: \[ C(75) = 75 + 15 = 90 \]

Step 3: Find C(90)

Since \( 90 < 100 \), we use the first part of the function: \[ C(90) = 90 + 15 = 105 \]

Step 4: Find C(100)

Since \( 100 \geq 100 \), we use the second part of the function: \[ C(100) = 100 \]

Step 5: Find C(105)

Since \( 105 \geq 100 \), we use the second part of the function: \[ C(105) = 105 \]

Step 6: Interpret the Results

The results show the total cost of purchasing books online for different book prices:- \( C(75) = 90 \): A \\(15 shipping fee applies when the book price is under \\)100.- \( C(90) = 105 \): A \\(15 shipping fee still applies under \\)100.- \( C(100) = 100 \) and \( C(105) = 105 \): No shipping fee when the purchase is \$100 or more.

Key Concepts

Function Evaluation

Evaluating a function is all about determining what value a function takes on given specific inputs. In this exercise, we have the cost function \( C(x) \) that involves piecewise conditions:

• For orders where the total price \( x \) is less than 100 dollars, the cost \( C(x) \) is the price plus a 15 dollar shipping fee.
• If the total price is 100 dollars or higher, the function simplifies as there is no shipping fee.

Function evaluation generally follows a few straightforward steps:

• Identify which part of the function to use based on the input.
• Substitute the input value into the chosen part of the function.
• Perform the calculations to find \( C(x) \).

By applying these steps, you can determine the costs for different book purchases, such as \( C(75) = 90 \), \( C(90) = 105 \), \( C(100) = 100 \), and \( C(105) = 105 \).

Step-by-Step Solution

Breaking down a problem into detailed steps can make solving it easier. For our function, this involves systematically evaluating \( C(x) \) for different values of \( x \). Here's how we approached it:

• Identify the function: Recognize that we have a piecewise function, which dictates different conditions for different scenarios based on the value of \( x \).
• Apply the conditions: Use associated conditions to decide whether the shipping fee applies based on the order value.
• Perform calculations: Execute the arithmetic based on the chosen part of the piecewise function. For instance, for \( C(75) \), we calculate \( 75 + 15 = 90 \) because \( 75 < 100 \).

Step-by-step solutions are helpful in visualizing the problem-solving process, ensuring a better understanding of function evaluations and applications.

Shipping Cost Analysis

This exercise serves as a practical example of shipping cost analysis using a piecewise function. The idea is to price shipping based on order value and thus incentivize larger purchases through free shipping offers. Here's how the analysis breaks down:

• Cost implementation: The bookstore uses a simple piecewise function to add a shipping charge of \( 15 \) dollars when the order total is below \( 100 \) dollars.
• Free shipping incentive: For orders \( 100 \) dollars or above, the shipping cost is zero, encouraging customers to purchase more books to save on shipping.
• Understanding the findings: By calculating \( C(75) \), \( C(90) \), \( C(100) \), and \( C(105) \), it's clear how the cost structure shifts based on purchase amount, highlighting the influence of free shipping on purchase decisions.

Analyzing shipping costs through piecewise functions demonstrates retail strategies in managing costs and promoting higher sales volumes effectively.