Problem 113 - College Algebra with Modeling and Visualization

Question

(Refer to Example 12.) A company charges \(\$ 20\) to make one monogrammed shirt, but reduces this cost by \(\$ 0.10\) per shirt for each additional shirt ordered up to 100 shirts. If the cost of an order is \(\$ 989,\) how many shirts were ordered?

Step-by-Step Solution

Answer

The company ordered 86 shirts.

Step 1: Define the problem

We know the base price for one shirt is $20, and the cost decreases by $0.10 sequentially for each additional shirt up to 100 shirts. We need to find out how many shirts were ordered if the total cost was $989.

Step 2: Establish variables and equation

Let \( x \) represent the number of shirts ordered. The cost for each additional shirt is given by the formula \( 20 - 0.10(x - 1) \). We then need to set up an equation for the total cost: \( x[20 - 0.10(x - 1)] = 989 \).

Step 3: Simplify the equation

Simplify the cost equation: \( 20x - 0.10x(x - 1) = 989 \). This expands to \(20x - 0.10x^2 + 0.10x = 989\).

Step 4: Consolidate terms

Combine like terms in the equation: \( 20.10x - 0.10x^2 = 989 \). Rearrange the equation in standard quadratic form: \[-0.10x^2 + 20.10x - 989 = 0.\]

Step 5: Solve the quadratic equation

We use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) with \( a = -0.10, \ b = 20.10, \ c = -989 \). Calculate the discriminant: \[b^2 - 4ac = (20.10)^2 - 4(-0.10)(-989).\]Continue with the calculation:\[b^2 - 4ac = 404.01 - 395.6 = 8.41.\]Now solve for \( x \): \[x = \frac{-20.10 \pm \sqrt{8.41}}{-0.20}.\]Calculate the square root and simplify: \[x = \frac{-20.10 \pm 2.9}{-0.20}.\]Thus, \[x = \frac{-20.10 + 2.9}{-0.20}\] and \ \[x = \frac{-20.10 - 2.9}{-0.20}.\]

Step 6: Calculate potential solutions

Compute each possibility:1. \( x = \frac{-17.20}{-0.20} = 86 \)2. \( x = \frac{-23.00}{-0.20} = 115 \)Since only orders up to 100 shirts can receive a discount, the valid solution for this context is \( x = 86 \).

Step 7: Verify solution

Check if \( x = 86 \) makes sense:Calculate total cost using the formula \( 20x - 0.10x(x - 1) \):\[20(86) - 0.10(86)(85) = 1720 - 731 = 989.\]This matches the original cost given. Thus, \( x = 86 \) is verified as correct.

Key Concepts

Problem Solving

Solving a problem begins by clearly identifying what is being asked. Here, we need to determine the number of shirts ordered if the total cost equates to $989. The problem involves understanding a series of relationships: the fixed cost of one shirt, the discount applied for each additional shirt, and the constraints provided (such as the maximum of 100 shirts receiving the discount).

Effective problem-solving often requires setting realistic goals and determining what information you already have:

• Identify known values: the initial shirt cost and the incremental decrease for each additional shirt.
• Clarify the unknowns: the number of shirts ordered.
• Consider any restrictions: a maximum discount applies only up to a certain number of items.

You then translate these factors into a workable equation or formula that allows you to solve for the unknown. Structured approaches like breaking the work into steps, as shown in the solution, ensure that no aspect of the problem is overlooked.

Algebraic Equations

Algebraic equations play a crucial role in translating problem statements into numerical forms that we can solve. In our exercise, we transformed the verbal description of monogrammed shirt pricing into a mathematical equation.

Here's a step-by-step on forming the relevant equation:

• Start with an expression for one shirt: the initial price minus the discount factor, which is a function of the number of shirts ordered.
• The general equation becomes: \( ext{Total Cost} = x[20 - 0.10(x - 1)] \).
• Simplify this expression to form a quadratic equation, enabling you to use the quadratic formula for finding the possible values of \( x \).

Understanding how to manipulate and form equations, including simplifying and rearranging terms, is essential. Quadratic equations, particularly, need a systematic approach: putting them into the standard form \( ax^2 + bx + c = 0 \) before solving through established methods such as factoring, completing the square, or using the quadratic formula.

Cost Analysis

Cost analysis involves understanding how changes in production or order quantities affect overall expenses. In this case, price alterations associated with bulk purchasing need to be precisely calculated to determine the number of items ordered.

Here’s how cost analysis applies in this scenario:

• The individual shirt cost decreases as more shirts are ordered, but only up to a certain limit (100 shirts in this exercise).
• Use the equation established in the problem-solving phase to calculate total expenditure based on the number of shirts.
• Verify calculations to ensure that the derived quantity both meets the given financial constraints and respects any rules about discounts or maximum limits on such reductions.

Performing these calculations demonstrates the essential nature of cost analysis — ensuring that businesses and consumers understand expenses and take advantage of cost reductions where applicable. This knowledge is vital in strategic planning and decision-making for optimal financial outcomes.