Problem 61 - Algebra 2 and Trigonometry

Question

The profit function, in thousands of dollars, for a company that makes graphing calculators is \(\mathrm{P}(x)=-5 x^{2}+5,400 x-106,000\) where \(x\) is the number of calculators sold in the millions. a. Graph the profit function \(\mathrm{P}(x)\) b. How many calculators must the company sell in order to make a profit?

Step-by-Step Solution

Answer

The company must sell more than 20.633 million calculators to make a profit.

Step 1: Understanding the Profit Function

The profit function is given as \( P(x) = -5x^2 + 5400x - 106000 \). This is a quadratic function of the form \( ax^2 + bx + c \), where \( a = -5 \), \( b = 5400 \), and \( c = -106000 \). It represents the profit, in thousands of dollars, based on the number of calculators sold, where \(x\) is in millions.

Step 2: Graphing the Profit Function

To graph this function, identify the vertex and intercepts. Since this is a downward-opening parabola (\( a < 0 \)), the vertex will be the maximum point. The vertex \((h, k)\) is found using \( h = -\frac{b}{2a} \), which gives \( h = -\frac{5400}{2 \times -5} = 540 \). Substitute \( x = 0 \) to find the y-intercept, \( P(0) = -106000 \), indicating when no calculators are sold, there is a loss of $106,000. Plot the function by calculating additional points or using a graphing tool.

Step 3: Finding the Break-Even Point

The company makes a profit when \( P(x) > 0 \), so set \( -5x^2 + 5400x - 106000 > 0 \). Solve \( -5x^2 + 5400x - 106000 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute \( a = -5 \), \( b = 5400 \), and \( c = -106000 \) to find the roots, which are the break-even points.

Step 4: Solving for Break-Even Points

Calculate the discriminant: \( \Delta = b^2 - 4ac = 5400^2 - 4 \times (-5) \times (-106000) = 29160000 - 2120000 = 27040000 \). Use the quadratic formula to find \( x = \frac{-5400 \pm \sqrt{27040000}}{-10} \), which simplifies to \( x \approx 20.633 \) and \( x \approx 1029.367 \). This means the company must sell between approximately 20.633 million and 1029.367 million calculators to make a profit.

Key Concepts

Profit Function

The profit function is a mathematical equation that calculates the profit a company earns based on various factors. For the company producing graphing calculators, the profit function is given as \( P(x) = -5x^2 + 5400x - 106000 \). This specific equation tells us:

• The term \( -5x^2 \) indicates a downward-opening parabola, representing a reduction in profit with every additional increase in the sales number.
• The linear term \( 5400x \) shows the increase in profit per calculator sold, while the constant \( -106000 \) depicts an initial loss when no calculators are sold. This is because of fixed costs or initial investments before any calculators are sold.

In simpler terms, understanding this equation helps to predict profitability based on the sales volume of graphing calculators, indicated in millions by variable \( x \). This can be a powerful tool for businesses to plan their production and financial strategies.

Graphing Quadratic Equations

Graphing quadratic equations involves plotting the curve that the equation represents on a coordinate plane. In our case, we are dealing with a quadratic function \( P(x) = -5x^2 + 5400x - 106000 \), which forms a parabola.To graph this equation properly:

• Identify key features like intercepts. The y-intercept is where the graph meets the y-axis, found by setting \( x = 0 \), giving us \( P(0) = -106000 \).
• The parabola's vertex is another critical point. It is found using the formula \( h = -\frac{b}{2a} \). For our equation, \( h = 540 \). This indicates the point of maximum profit as the parabola opens downwards.
• Using a graphing tool, plot this vertex to visualize how the profit changes over different sales volumes. Use additional points to define the shape of the parabola if necessary.

Graphing makes the understanding of profit compared to sales numbers much easier and clearer, especially in visualizing the peak profit points and overall trends.

Break-Even Analysis

Break-even analysis is a method used to determine when an investment will start generating profit. For the graphing calculator company, the break-even point is the sales volume where profit starts happening, represented by \( P(x) > 0 \).Expanding on our function:

• Set the profit equation to zero: \( -5x^2 + 5400x - 106000 = 0 \).
• Apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the values of \( x \), which are the solutions to when the profit is zero. These solutions are the break-even points.
• The discriminant was calculated as \( \Delta = 27040000 \), and solved \( x \) values are approximately 20.633 million and 1029.367 million calculators sold.

This analysis shows that the company starts making a profit once they sell more than around 20.633 million calculators and up to around 1029.367 million, meaning early sales cover fixed costs and consequent sales increase profits.

Vertex of a Parabola

The vertex of a parabola is a pivotal point that gives the maximum or minimum value of the quadratic function, depending on the function's orientation. In our downward-opening parabola \( P(x) = -5x^2 + 5400x - 106000 \), the vertex represents the point of maximum profit.How to find the vertex:

• Utilize the formula \( h = -\frac{b}{2a} \) to find the x-coordinate of the vertex. For this profit function, \( h = 540 \), meaning it occurs when 540 million calculators are sold.
• Substitute \( x = 540 \) back into the profit equation to find the maximum profit value, \( k = P(540) \). This gives the highest profit point.

The vertex is crucial as it helps the company understand under what conditions they achieve the best possible profit. This understanding can guide future sales and marketing strategies to aim towards that ideal sales volume.