Problem 58 - College Algebra

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 21 and 93 phones each day for profit.

Step 1: Define the Profit Function

To determine the profit, we need to find the profit function, which is the revenue function minus the cost function. Let's denote the profit function by \( P(x) \). Thus, \( P(x) = R(x) - C(x) \).

Step 2: Substitute the Given Functions

Let’s substitute the revenue function \( R(x) = -3x^2 + 480x \) and the cost function \( C(x) = 8x^2 - 600x + 21,500 \) into the profit function. This gives:\[ P(x) = (-3x^2 + 480x) - (8x^2 - 600x + 21,500) \].

Step 3: Simplify the Profit Function

Now, simplify the profit function by distributing the negative sign and combining like terms:\[P(x) = -3x^2 + 480x - 8x^2 + 600x - 21,500 = -11x^2 + 1080x - 21,500\].

Step 4: Set Up the Inequality for Profit

For the company to generate a profit, the profit function must be greater than zero:\[-11x^2 + 1080x - 21,500 > 0 \].

Step 5: Solve the Inequality

To find the range of \( x \) that satisfies this inequality, solve \(-11x^2 + 1080x - 21,500 = 0\) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \], where \(a = -11\), \(b = 1080\) and \(c = -21,500\).

Step 6: Calculate the Discriminant

First, calculate the discriminant:\[b^2 - 4ac = 1080^2 - 4(-11)(-21,500) \].Compute this value to find whether there are real solutions.

Step 7: Compute the Roots

Now compute the roots using the quadratic formula:Compute \(x = \frac{-1080 \pm \sqrt{1080^2 - 4(-11)(-21,500)}}{2(-11)}\) to find the points at which profit is zero.

Step 8: Determine the Profit Range

Evaluate the roots for the inequality and determine the range of \(x\) such that the profit is positive. This is the open interval between the roots.

Step 9: Round to Nearest Production Number

Round the solution to the nearest integer to find the minimum and maximum number of phones that need to be produced for profit.

Key Concepts

Profit Function

The profit function is a fundamental concept in economics and business. It helps businesses understand how profitable they are at different levels of production. Simply put, the profit function is calculated by subtracting the cost function from the revenue function. In mathematical terms, given the revenue function as \(R(x)\) and the cost function as \(C(x)\), the profit function \(P(x)\) is derived as:

\[P(x) = R(x) - C(x)\]

This equation shows us the net income from producing "x" number of items. It is important because it informs the decision-makers about the levels of production that would result in the best profit. The value of \(x\) where \(P(x) > 0\) indicates that the company is making money after covering all their costs.

Revenue Function

Revenue function is the expression that predicts how much money a company makes from selling a certain number of goods or services. If the company solely sells cell phones, the revenue function, denoted as \(R(x)\), would represent income from "x" phones sold.

In terms of our given exercise, the revenue function \(R(x) = -3x^2 + 480x\) shows a quadratic behavior, meaning the revenue made per unit begins to decrease after a certain point. This uncommon pattern might occur due to factors like increased competition or market saturation, where producing too many units reduces each unit's value. Identifying this optimal point is crucial for maximizing revenue.

Cost Function

The cost function captures all the various expenses incurred while producing a particular number of goods. These costs can include materials, labor, overhead, and other operational elements. From the analysis perspective, it’s denoted as \(C(x)\) and is imperative to understanding the financial health of a business.

In our example, the cost function is \(C(x) = 8x^2 - 600x + 21,500\). This suggests a quadratic cost model, indicating that costs don't increase linearly with production. Uncovering cost behaviors, like economies of scale or inefficiencies, can help optimize the number of cell phones produced to a level where excess costs don’t cancel out potential profits.

Quadratic Formula

The quadratic formula is a powerful tool used in solving quadratic equations. These are equations of the form \(ax^2 + bx + c = 0\). For our exercise, where we solve the profit function to determine profitable production levels, the quadratic formula allows us to find the values of \(x\) where the profit function \(P(x) = 0\).

The quadratic formula is expressed as:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Here, \(a\), \(b\), and \(c\) correspond to the coefficients from the equation. By calculating the "discriminant" \(b^2 - 4ac\), we determine the nature of the roots (real or imaginary). For profit calculations, real roots highlight those crucial production points where revenue precisely balances costs, indicating zero profit points. Knowing where these points are helps businesses set a range where profit is positive and prioritizes efficient resource management.