Problem 57 - 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 laptop company has discovered their cost and revenue functions for each day: \(C(x)=3 x^{2}-10 x+200\) and \(R(x)=-2 x^{2}+100 x+50\). If they want to make a profit, what is the range of laptops per day that they should produce? Round to the nearest number which would generate profit.

Step-by-Step Solution

Answer

Produce between 2 and 20 laptops per day for profit.

Step 1: Understand the Problem

The problem requires us to find the range of laptops (x) that must be produced each day for the company to make a profit. Profit is the difference between revenue and cost.

Step 2: Define the Profit Function

The profit function \( P(x) \) is given by the difference between the revenue function \( R(x) \) and the cost function \( C(x) \): \[ P(x) = R(x) - C(x) \]Substituting the given functions:\[ P(x) = (-2x^2 + 100x + 50) - (3x^2 - 10x + 200) \]

Step 3: Simplify the Profit Function

Combine like terms to simplify the profit equation:\[ P(x) = -2x^2 + 100x + 50 - 3x^2 + 10x - 200 \]Results in:\[ P(x) = -5x^2 + 110x - 150 \]

Step 4: Solve for Profit Greater than Zero

To find the range of laptops that generate profit, solve for when the profit function is greater than zero:\[ P(x) > 0 \Rightarrow -5x^2 + 110x - 150 > 0 \]

Step 5: Solve the Inequality

Solve the inequality \(-5x^2 + 110x - 150 = 0\) to find the points where the profit is zero. Use the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a = -5\), \(b = 110\), \(c = -150\).

Step 6: Calculate the Solutions Using Quadratic Formula

Calculate the discriminant:\[ \Delta = 110^2 - 4 \cdot (-5) \cdot (-150) = 12100 - 3000 = 9100 \]Substitute back into the quadratic formula:\[ x = \frac{-110 \pm \sqrt{9100}}{-10} \]Approximate \( \sqrt{9100} \approx 95.4 \), so:\[ x = \frac{-110 + 95.4}{-10}, \frac{-110 - 95.4}{-10} \]

Step 7: Simplify the Results

Calculate the approximate values:\[ x = \frac{-14.6}{-10} = 1.46 \] and \[ x = \frac{-205.4}{-10} = 20.54 \]Thus, the interval where profit occurs is between these values.

Step 8: Determine the Range of Laptops

Round the solutions to the nearest whole number. The company should produce between 2 and 20 laptops per day to make a profit.

Key Concepts

Profit Function

The concept of a profit function is fundamental in business and economics. It helps in understanding how a company's financial performance is linked to its operations.
To calculate profit, you subtract the cost from revenue:

• Revenue Function, \(R(x)\): It's the amount of money the company brings in from sales.
• Cost Function, \(C(x)\): It's the total expense incurred to produce goods or services.
• Profit Function, \(P(x)\): It is determined by \(P(x) = R(x) - C(x)\).

In our exercise, the profit function is used to represent and solve for the company's profitability based on the number of laptops produced.
The importance of determining this is to maximize profitability by finding the optimal range of production.

Quadratic Formula

The quadratic formula is a mathematical tool used to solve quadratic equations, which are polynomials of the form \(ax^2 + bx + c = 0\).
It is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\]

• \(a\), \(b\), and \(c\) are coefficients of the equation.
• The term \(b^2 - 4ac\) is known as the discriminant.
• The discriminant tells us about the nature of the roots:
• If positive, the equation has two distinct real roots.
• If zero, it has exactly one real root.
• If negative, no real roots exist.

In the exercise, the quadratic formula helped find when profit was zero. These points mark the boundaries of production range for profitability.

Inequalities

Inequalities are used to describe a range of possible solutions rather than precise values. They are essential in optimizing decisions, like determining profitable production levels.
In mathematical terms, solving inequalities informs us about intervals where a function's output meets specific conditions:

• Signifies greater than (>) when profit is more than zero.
• Signifies less than (<) or equal conditions for non-profit scenarios.

For this exercise:

• The inequality \(-5x^2 + 110x - 150 > 0\) provides the range of \(x\) where profit is signaled.
• Solving it gives the values between which producing laptops is financially beneficial.
• This is crucial data for the company to determine ideal production levels for profitability.

By understanding inequalities, businesses can make informed decisions about their operations to ensure they stay in the black.