Problem 36 - Beginning and Intermediate Algebra

Question

Write a system of equations and solve. A backpack manufacturer estimates that the cost \(y,\) in dollars, to make \(x\) thousands of backpacks is given by $$ y=9 x^{2}+30 x+18 $$ The revenue \(y,\) in dollars, from the sale of \(x\) thousands of backpacks is given by $$ y=21 x^{2} $$ Find the break-even point for the manufacture and sale of the backpacks. (See problem 35 for explanation.)

Step-by-Step Solution

Answer

The break-even point for the manufacture and sale of the backpacks is reached when 3,000 backpacks are produced and sold. This is found by setting the cost function (\( y = 9x^2 + 30x + 18 \)) equal to the revenue function (\( y = 21x^2 \)) and solving for x, which yields two solutions: \( x = -\frac{1}{2} \) and \( x = 3 \). Since the first solution is not valid in the context of the problem, the break-even point is at \( x = 3 \), or 3,000 backpacks.

Step 1: Set up the system of equations

To find the break-even point, we need to set the cost function equal to the revenue function. In this case, we have: \( 9x^2 + 30x + 18 = 21x^2 \)

Step 2: Solve for x

To solve for x, we'll first subtract the cost function from the revenue function to get a quadratic equation. Then, we'll solve the quadratic equation for the possible break-even points. Subtract \(9x^2\) from both sides: \( 30x + 18 = 12x^2 \) Now, we'll rearrange the equation by subtracting the left side from the right side, to get: \( 12x^2 - 30x - 18 = 0 \)

Step 3: Solve the quadratic equation

Now, we need to find the values of x that satisfy the equation above. In order to solve this equation, we can either use the quadratic formula, factoring, or completing the square. In this case, the quadratic equation can be factored. We can factor by dividing all terms by 6: \( 2x^2 - 5x - 3 = 0 \) Now, we can factor the quadratic equation: \( (2x + 1)(x - 3) = 0 \) From this factored form, we can find two possible solutions for x: 1. \( 2x + 1 = 0 => x = -\frac{1}{2} \) 2. \( x - 3 = 0 => x = 3 \)

Step 4: Evaluate the solutions

The first solution, x = -0.5, does not make sense in the context of this problem since we cannot produce negative thousands of backpacks. Therefore, we reject this solution. The second solution, x = 3, is a valid solution. This means that at the production and sale of 3,000 backpacks, the break-even point is reached. So, the break-even point for the manufacture and sale of the backpacks is when 3,000 backpacks are produced and sold.

Key Concepts

Quadratic Equations

Quadratic equations are polynomials that have a squared term as their highest exponent. They are typically written in the standard form:

• \( ax^2 + bx + c = 0 \)

Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). The solutions to these equations are the values of \( x \) that satisfy the equation. Quadratic equations can be solved using several methods:

• The quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Factoring
• Completing the square

In our exercise, we use factoring to find the solution to the quadratic equation that arises when determining the break-even point.

Understanding how to solve quadratic equations is crucial for break-even analysis, especially when dealing with cost and revenue functions that are represented as quadratic equations.

Cost Function

A cost function is a mathematical expression that describes the total cost incurred by producing a certain number of goods. It is often represented in terms of variables like \( x \), which indicates the quantity of goods produced. For instance, the cost function given in the exercise is:

• \( y = 9x^2 + 30x + 18 \)

This quadratic function suggests that as production increases, costs rise, but not in a purely linear fashion. Instead, the squared term \( 9x^2 \) indicates that costs accelerate as more backpacks are produced. This is typical of many manufacturing processes where additional production requires more significant investments in skills, materials, or technology.

Understanding a cost function is vital for businesses. It helps them predict expenses and make informed production decisions, like determining pricing strategies or identifying the optimal production level to maximize profit.

Revenue Function

A revenue function expresses the total revenue generated from selling a certain number of items. Revenue functions are crucial for predicting income and assessing business performance. In our example, the revenue function is:

• \( y = 21x^2 \)

Here, \( x \) denotes thousands of backpacks sold, while \( 21 \) represents the price per backpack or the revenue generated per unit. This quadratic revenue function suggests an increase in revenue as more items are sold, similar to a parabolic increase.

This pattern of increasing revenue with more sales reflects economies of scale, where larger production can lead to greater revenue. Understanding the revenue function allows businesses to develop strategies aimed at maximizing profits, such as pricing adjustments or sales targets.

Factoring Quadratics

Factoring quadratics is a method used to solve quadratic equations, making it a valuable technique in solving break-even problems. When you factor a quadratic equation, you express it as the product of two binomials. This is what we did in the exercise:

• \( 12x^2 - 30x - 18 = 0 \)
• Factored into: \( (2x + 1)(x - 3) = 0 \)

Each of the terms in parentheses can be set to zero, giving potential solutions for \( x \). Factoring is preferable when the quadratic expression can be easily decomposed into simpler components.

This method is efficient and often faster than using the quadratic formula or completing the square, especially when the numbers and coefficients involved are manageable. Factoring helps us identify the values of \( x \) that bring revenue and costs to equilibrium, thus solving for the break-even point where business neither profits nor loses.