Problem 76 - Elementary Algebra

Question

A manufacturing company has determined that the daily revenue, \(R\), in thousands of dollars depends on the number, \(n\), of palettes of product sold according to the formula \(R=12 n-0.6 n 2 .\) Determine the number of palettes that must be sold in order to maintain revenues at \(\$ 60,000\) per day.

Step-by-Step Solution

Answer

The company must sell 10 palettes to meet the revenue target of $60,000.

Step 1: Understand the equation

The revenue equation given is \( R = 12n - 0.6n^2 \). Here, \( R \) represents the revenue in thousands of dollars, and \( n \) is the number of palettes sold.

Step 2: Set up the equation

To find the number of palettes needed to reach \( \$60,000 \), first convert the target revenue from dollars to thousands of dollars: \( 60,000 \) dollars is \( 60 \) thousand dollars. So, set up the equation: \( 12n - 0.6n^2 = 60 \).

Step 3: Rearrange into a standard form

Rearrange the equation into a standard quadratic equation form: \( -0.6n^2 + 12n - 60 = 0 \).

Step 4: Simplify the equation

To simplify, multiply the entire equation by \( -10 \) to avoid decimals: \( 6n^2 - 120n + 600 = 0 \).

Step 5: Use the quadratic formula

The quadratic formula is \( n = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \). Identify \( a = 6 \), \( b = -120 \), and \( c = 600 \). Substitute into the formula: \( n = \frac{{120 \pm \sqrt{{(-120)^2 - 4 \times 6 \times 600}}}}{{2 \times 6}} \).

Step 6: Calculate the discriminant

Calculate the discriminant: \( (-120)^2 - 4 \times 6 \times 600 = 14400 - 14400 = 0 \). The discriminant is 0, indicating one solution.

Step 7: Solve for n

Since the discriminant is 0, use the quadratic formula to find: \( n = \frac{{120}}{{12}} = 10 \).

Step 8: Interpret the result

The solution \( n = 10 \) means the company must sell 10 palettes of product for a daily revenue of \( \$60,000 \).

Key Concepts

Revenue Calculation

Understanding how a company's revenue depends on the number of items sold is essential in business operations. Revenue calculation allows a business to predict earnings based on sales figures. In our scenario, the company uses a mathematical equation to represent this relationship:

• Given is the formula: \( R = 12n - 0.6n^2 \), where \( R \) is the revenue in thousands of dollars, and \( n \) is the number of palettes sold.

This quadratic equation models how revenue faces initial growth with sales but tends to decrease after reaching a peak. This drop occurs due to the \( -0.6n^2 \) term which represents diminishing returns as more palettes are sold. For a specific sales target, such as achieving \( \$60,000 \) revenue, you substitute \( 60 \) (since revenue is in thousand dollars) into the equation and solve for \( n \). This process provides insights into the quantity required to hit specific financial goals efficiently.

Quadratic Formula

The quadratic formula is a robust tool for solving quadratic equations, which are equations in the form \( ax^2 + bx + c = 0 \). This formula provides solutions for \( x \) based on the coefficients \( a \), \( b \), and \( c \). The quadratic formula is depicted as:

• \( n = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \)

In our exercise, we identified \( a = 6 \), \( b = -120 \), and \( c = 600 \) from the rearranged revenue equation. By substituting these values into the formula, you can solve for \( n \), giving the number of palettes needed to achieve the desired revenue. This approach is highly reliable for problems where straightforward factoring is complex or infeasible. Mastering the quadratic formula equips you to handle a variety of mathematical scenarios beyond business contexts.

Discriminant

The discriminant is a component of the quadratic formula under the square root symbol, noted as \( b^2 - 4ac \). It tells us about the nature and number of solutions a quadratic equation has:

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is exactly one real solution (it is a perfect square).
• If negative, no real solutions exist, only complex numbers.

In the context of our exercise, the discriminant calculated as \( 14400 - 14400 = 0 \). A zero discriminant indicates there is exactly one solution. This means only one specific quantity of palettes sold will generate exactly \( \$60,000 \) in revenue. Understanding the discriminant greatly aids in predicting the outcome of quadratic relationships, providing valuable decision-making information in both academic and real-world settings.