Problem 84 - College Algebra with Modeling and Visualization

Question

cost of Carpet Each foot of carpet purchased from a 12-foot-wide roll costs \(\$ 36 .\) If a fraction of a foot is purchased, a customer does not pay for the extra amount. For example, if a customer wants 14 feet of carpet and the salesperson cuts off 14 feet 4 inches, the customer does not pay for the extra 4 inches. (a) How much does 9 feet 8 inches of carpet from this roll cost? (b) Using the greatest integer function, write a formula for the price \(P\) of \(x\) feet of carpet.

Step-by-Step Solution

Answer

(a) \$324; (b) \(P(x) = 36 \times \lfloor x \rfloor\).

Step 1: Understanding the Problem

We need to determine the cost of a carpet that is 9 feet 8 inches in length, knowing that 1 foot of carpet costs \$36. The key point to note is that the customer doesn't pay for partial feet of carpet. Then we'll derive a formula for the cost of \(x\) feet of carpet using the greatest integer function.

Step 2: Convert Inches to Feet

First, let's convert the entire length of the carpet from feet and inches to feet. Since 1 foot equals 12 inches, 8 inches is a fraction of a foot calculated as \(\frac{8}{12} = \frac{2}{3}\) of a foot.

Step 3: Calculate Total Feet

Add the converted fraction of a foot to the whole feet to find the total length in feet: \(9 + \frac{2}{3} = \frac{29}{3}\) feet.

Step 4: Apply Greatest Integer Function

The customer only pays for whole feet. Therefore, we apply the greatest integer function, also known as the floor function, which is the greatest integer less than or equal to a given number. \(\lfloor \frac{29}{3} \rfloor = 9\).

Step 5: Calculate Total Cost for Given Length

Since customers only pay for whole feet, and each foot costs \\(36: \(9 \text{ feet} \times 36 \text{ per foot} = 324\). Thus, the cost of 9 feet 8 inches is \\)324.

Step 6: Derive the Formula for Cost

For any feet \(x\), we use the greatest integer function to calculate the number of whole feet a customer will pay for: \(P(x) = 36 \times \lfloor x \rfloor\).

Key Concepts

Understanding the Floor Function

The floor function, commonly referred to as the greatest integer function, is a mathematical operation defined as the greatest integer less than or equal to a given number. It's represented by the symbol \(\lfloor x \rfloor\). For example, the floor of 5.9 is 5, and the floor of -2.1 is -3. The floor function is very useful when dealing with real-world measurements or transactions where only whole quantities are used.

In the context of our carpet example, the floor function helps determine how many complete feet of carpet a customer must pay for. Even if a customer requires slightly more than a whole number of feet, only the whole feet are considered for billing. By applying this function, we ensure that customers are not charged for fractions of a foot that they do not fully receive.

Simplifying Cost Calculation

Cost calculation can become straightforward with the use of the floor function. In our exercise, carpet costs are determined per whole foot. Each foot costs $36, and the floor function helps us calculate the number of payable whole feet.

Here's a quick breakdown:

• Convert the measurement (feet and inches) into feet. For example, 9 feet 8 inches becomes \(9 + \frac{2}{3} = \frac{29}{3}\) feet.
• Apply the floor function: \(\lfloor \frac{29}{3} \rfloor = 9\) feet.
• Multiply the total payable feet by the cost per foot: \(9 \times 36 = 324\).

This simplifies the billing process and ensures consistency, as customers pay only for complete units, consistent with service policies.

The Role of Mathematical Modeling

Mathematical modeling is a powerful tool in creating representations of real-world scenarios using mathematical concepts and language. In this exercise, mathematical modeling enables us to construct a formula that accurately calculates the cost of carpet based on the length requested by a customer.

We derive the formula \(P(x) = 36 \times \lfloor x \rfloor\), where \(P(x)\) represents the price, and \(x\) is the length in feet. This model captures the essential service policy that customers pay only for completely usable feet of carpet. Using this formula, any customer query regarding carpet pricing for a specified length can quickly be resolved.

Mathematical modeling provides clarity and precision to decision-making processes in business contexts, ensuring that all parties understand the pricing structure clearly and in advance.