Problem 58 - College Algebra with Corequisite Support

Question

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions. A cell phone company has the following cost and revenue functions: \(C(x)=8 x^{2}-600 x+21,500\) and \(R(x)=-3 x^{2}+480 x .\) What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

Step-by-Step Solution

Answer

Produce between 28 and 73 cell phones per day for profit.

Step 1: Define the Profit Function

The profit is the revenue minus the cost, which can be represented as \( P(x) = R(x) - C(x) \). For the given functions, the profit equation is:\[ P(x) = (-3x^2 + 480x) - (8x^2 - 600x + 21,500) \]Simplifying the equation, we have:\[ P(x) = -3x^2 + 480x - 8x^2 + 600x - 21,500 \]\[ P(x) = -11x^2 + 1080x - 21,500 \]

Step 2: Set the Profit Function Greater Than Zero

To determine when the company makes a profit, set the profit function greater than zero:\[ -11x^2 + 1080x - 21,500 > 0 \]This inequality represents the range of \( x \) values where profit exists.

Step 3: Find the Roots of the Quadratic

Solve the equation \( -11x^2 + 1080x - 21,500 = 0 \) to find the critical points using the quadratic formula, where \( a = -11 \), \( b = 1080 \), and \( c = -21500 \):\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]First, compute the discriminant:\[ b^2 - 4ac = 1080^2 - 4(-11)(-21,500) \]\[ = 1,166,400 - 946,000 \]\[ = 220,400 \]Now find the roots:\[ x = \frac{-1080 \pm \sqrt{220,400}}{-22} \]\[ x = \frac{-1080 \pm 469.20}{-22} \]Solving this, the roots are approximately:\( x_1 = 27 \) and \( x_2 = 73.6 \).

Step 4: Determine the Range for Profit

Since the profit function is a downward-opening parabola (coefficient of \( x^2 \) is negative), the company makes a profit between the two roots. Thus, the range of production is \( 27 < x < 74 \).

Step 5: Conclusion

To ensure profit, the company should produce between 28 and 73 cell phones per day, rounding the root values to the nearest integers that satisfy profit conditions.

Key Concepts

Profit Function

In the context of a cell phone company, the profit function represents the difference between the revenue generated from selling cell phones and the costs incurred in producing them. This is crucial for determining the financial health of the business. The profit function is mathematically defined as:

• Profit = Revenue - Cost

For this exercise, we've defined the profit function as: \[ P(x) = R(x) - C(x) \] After plugging in the given cost \( C(x) \) and revenue \( R(x) \) functions, we simplify it to:\[ P(x) = -11x^2 + 1080x - 21,500 \] The objective is to find out when this profit function is positive—which tells us the range of production that generates a profit. A positive value of \( P(x) \) indicates that revenue exceeds costs, granting a financial gain.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation. A quadratic equation is in the form of \( ax^2 + bx + c = 0 \). For our exercise, the simplified profit function is represented as \[ -11x^2 + 1080x - 21,500 = 0 \]. To find where this function is zero, we apply the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula helps in calculating the roots, which are the critical points where the function changes behavior. In our problem:

• The discriminant \( b^2 - 4ac \) is calculated first. This tells us the nature of roots:
  • If positive, there are two distinct real roots.

The roots define the intervals where the profit function changes from negative to positive values.

Cost and Revenue Functions

The construction of cost and revenue functions is essential in any business model. These functions help visualize and calculate earnings and expenses over a given number of goods produced or sold. In this cell phone company scenario:

• The cost function \( C(x) = 8x^2 - 600x + 21,500 \) accounts for the variable and fixed costs related to producing cell phones.
• The revenue function \( R(x) = -3x^2 + 480x \) reflects the income from selling cell phones.

Both functions, when examined, reveal how expenses vary with production volume and how revenue fluctuates with sales quantity. Understanding these is essential for pinpointing profit-maximizing production levels.

Inequalities in Algebra

Inequalities are a fundamental part of solving problems about practical domains like profit calculations. In this exercise, once we've formulated the profit function, we need to establish when it is greater than zero:\[ -11x^2 + 1080x - 21,500 > 0 \] This inequality tells us when the profit function is positive, indicating profitable production levels.To solve this, it involves:

• Finding the roots of the equation using the quadratic formula.
• Analyzing the intervals between roots to determine when the inequality holds true.

In practice, such inequalities help identify the most efficient production levels, ensuring the business operates within profitable bounds. Understanding this keeps the company aware of operational limits to ensure maximum profitability.