Problem 55 - College Algebra with Corequisite Support

Question

The cost function for a certain company is \(C=60 x+300\) and the revenue is given by \(R=100 x-0.5 x^{2}\). Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of \(x\) (production level) that will create a profit of $$\$ 300.$$

Step-by-Step Solution

Answer

The production levels are 20 and 60.

Step 1: Write the Profit Function

Profit is calculated by subtracting the cost function from the revenue function. Given, \( R = 100x - 0.5x^2 \) and \( C = 60x + 300 \), the profit \( P \) is \( P = R - C = (100x - 0.5x^2) - (60x + 300) \). Simplifying, \( P = -0.5x^2 + 40x - 300 \).

Step 2: Set Profit to $300

To find the production levels that yield a profit of $300, set the profit function equal to 300: \( -0.5x^2 + 40x - 300 = 300 \).

Step 3: Rearrange to Standard Quadratic Form

Rearrange the equation from Step 2 to the standard quadratic form: \( -0.5x^2 + 40x - 300 - 300 = 0 \), which simplifies to \( -0.5x^2 + 40x - 600 = 0 \).

Step 4: Clear the Coefficient of the Quadratic Term

Multiply through by \(-2\) to eliminate the decimal: \( x^2 - 80x + 1200 = 0 \). This is the quadratic equation to solve.

Step 5: Use the Quadratic Formula

Apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( x \). Here, \( a = 1 \), \( b = -80 \), and \( c = 1200 \).

Step 6: Calculate the Discriminant

Calculate the discriminant: \( b^2 - 4ac = (-80)^2 - 4(1)(1200) = 6400 - 4800 = 1600 \).

Step 7: Solve for x

Substitute the discriminant into the quadratic formula: \( x = \frac{80 \pm \sqrt{1600}}{2} \). Since \( \sqrt{1600} = 40 \), solve to get \( x = \frac{80 \pm 40}{2} \).

Step 8: Determine Both Values of x

Calculate the two possible values for \( x \): \( x = \frac{80 + 40}{2} = 60 \) and \( x = \frac{80 - 40}{2} = 20 \). Therefore, the production levels that yield a profit of $300 are \( x = 60 \) and \( x = 20 \).

Key Concepts

Profit Function

A profit function is a formula that helps calculate the profit a business can expect based on the revenue and cost functions. It's like a mathematical recipe to determine how much money a company keeps after covering its expenses. In simple terms, the profit function is derived by subtracting the cost function from the revenue function.
For example, given a revenue function of \( R = 100x - 0.5x^2 \) and a cost function of \( C = 60x + 300 \), the profit function would be \( P = R - C = (100x - 0.5x^2) - (60x + 300) \). Simplifying this expression gives you \( P = -0.5x^2 + 40x - 300 \).
Understanding the profit function is crucial because it allows you to see how different levels of production affect your net earnings. By setting this function to a specific profit amount, such as $300 in this example, you can solve for the production levels \( x \) that meet your financial goals.

Cost Function

The cost function represents the total cost incurred by a company to produce a certain level of goods. It usually comprises fixed costs and variable costs. Fixed costs are expenses that do not change with the level of production, like rent or salaries. Variable costs, on the other hand, change with the amount of goods produced.
In our exercise, the cost function is given by \( C = 60x + 300 \), where \( 60x \) represents the variable cost depending on the production level \( x \), and 300 is the fixed cost component.
Understanding the cost function is critical for businesses to ensure that they can cover these expenses while planning for profit. It lays the groundwork for setting prices and understanding the level of production necessary to break even and eventually earn a profit.

Revenue Function

The revenue function calculates the total income the business generates from selling its products. It is typically expressed as a function of the quantity of items sold. This function plays a pivotal role in determining how well a company is doing financially.
For instance, in the given exercise, the revenue function is \( R = 100x - 0.5x^2 \), which implies that revenue depends not only linearly on the quantity of products \( x \) but also includes a quadratic term \(-0.5x^2\). The quadratic term suggests that there may be a diminishing increase in revenue with increased production, possibly due to factors like market saturation or increased competition.
Understanding the revenue function helps businesses in forecasting potential income and adjusting production strategies to maximize earnings. By analyzing changes to this function, one can make informed decisions on the most profitable levels of output.