Output Level For TR=100: 85q-2q^2 Revenue Analysis

Let's dive into the world of revenue functions! In this article, we'll explore how to determine the output level (q) at which the total revenue (TR) reaches a specific target, in this case, 100. We'll use the given function TR = 85q - 2q^2 and walk through the steps to solve this problem. Understanding revenue functions is crucial for businesses to make informed decisions about production and pricing strategies. So, let's get started!

Understanding the Total Revenue Function

The total revenue function, TR = 85q - 2q^2, is a mathematical representation of how much revenue a business generates based on the quantity of goods or services it sells. In this equation:

• TR stands for Total Revenue, which is the total income received from selling a certain quantity.
• q represents the quantity of output, meaning the number of units sold.
• 85q indicates that for each unit sold, the revenue increases by 85 units of currency (e.g., dollars).
• -2q^2 represents a quadratic term, which suggests that as the quantity increases, the rate of revenue increase slows down and eventually may decrease. This is common in real-world scenarios due to factors like market saturation or decreasing prices at higher quantities.

The function is a quadratic equation, which means its graph is a parabola. The shape of the parabola tells us a lot about the revenue pattern. Since the coefficient of the q^2 term is negative (-2), the parabola opens downwards. This implies that there's a maximum point on the curve, representing the output level that maximizes total revenue. Understanding this function allows us to analyze how changes in output affect the total revenue, helping businesses optimize their production levels. Analyzing the components of the total revenue function, 85q and -2q^2, reveals key insights into the relationship between quantity and revenue. The linear term, 85q, indicates a direct, positive relationship, meaning revenue increases linearly with each additional unit sold. However, the quadratic term, -2q^2, introduces a non-linear element, suggesting diminishing returns as quantity increases. This is a common phenomenon in business, where the marginal revenue (the additional revenue from selling one more unit) decreases as production volume rises. This could be due to market saturation, increased competition, or the need to lower prices to sell larger quantities. The negative coefficient (-2) signifies that the parabola opens downwards, implying that there is a peak point representing the maximum revenue. Understanding this interplay between the linear and quadratic terms is crucial for businesses to make informed decisions about production levels and pricing strategies. For instance, if a company focuses solely on increasing quantity without considering the impact of the quadratic term, it might produce beyond the optimal level, leading to decreased overall revenue. Therefore, businesses need to carefully analyze their total revenue function to identify the point where marginal revenue equals marginal cost, maximizing their profitability. To truly understand the dynamics of revenue generation, it's essential to grasp the implications of both the linear and quadratic components within the total revenue function.

Setting up the Equation

Our goal is to find the output level (q) when the total revenue (TR) is 100. To do this, we need to set up the equation by substituting TR with 100 in the given function:

100 = 85q - 2q^2

This equation is a quadratic equation, and to solve it, we need to rearrange it into the standard quadratic form, which is:

ax^2 + bx + c = 0

In our case, q is the variable, and we need to rearrange the equation to match the standard form. Let's do that now.

To rearrange the equation 100 = 85q - 2q^2 into the standard quadratic form (ax^2 + bx + c = 0), we need to move all terms to one side of the equation, leaving zero on the other side. This process involves adding 2q^2 and subtracting 85q from both sides of the equation. This will ensure that the quadratic term has a positive coefficient, which simplifies the subsequent steps in solving the equation. The standard form is essential for applying various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Each of these methods relies on the equation being in the standard form to ensure accurate results. By rearranging the equation into standard form, we create a clear and structured representation of the problem, making it easier to identify the coefficients a, b, and c, which are crucial for applying the quadratic formula. The standard form also provides a visual representation of the parabolic nature of the equation, helping to understand the relationship between the variable q and the total revenue. This visual understanding can be valuable in interpreting the solutions and their implications in a business context. For instance, it can help identify the region where the total revenue is positive and the points where the revenue is equal to zero. The rearranged equation in standard form provides a solid foundation for further analysis and problem-solving.

Rearranging into Standard Quadratic Form

To get the equation into standard form, we'll add 2q^2 to both sides and subtract 85q from both sides:

2q^2 - 85q + 100 = 0

Now, our equation is in the standard quadratic form, where:

• a = 2
• b = -85
• c = 100

With the equation in this form, we can use several methods to solve for q. The most common methods are factoring, completing the square, and using the quadratic formula. Let's explore the quadratic formula, as it's a reliable method for solving any quadratic equation.

Having successfully rearranged the equation into standard quadratic form, we are now equipped with a clear understanding of the coefficients: a = 2, b = -85, and c = 100. These coefficients are the key ingredients for applying various methods to solve for q, representing the quantity at which the total revenue equals 100. While factoring can be a quick method for certain quadratic equations, it is not always straightforward, particularly when the coefficients are large or the roots are not integers. Completing the square is another method that can be used, but it can be more cumbersome and prone to errors compared to the quadratic formula. The quadratic formula stands out as a versatile and reliable method for solving quadratic equations, regardless of the complexity of the coefficients or the nature of the roots. Its standardized approach ensures that we can find the solutions efficiently and accurately. Furthermore, the quadratic formula provides valuable insights into the nature of the roots, revealing whether they are real or complex, and whether there are one, two, or no real solutions. This information is crucial for interpreting the results in the context of the problem. For example, if the discriminant (b^2 - 4ac) is negative, there are no real solutions, indicating that there is no output level at which the total revenue is exactly 100. Therefore, by identifying the coefficients in the standard form, we have paved the way for a systematic and robust solution using the quadratic formula.

Solving with the Quadratic Formula

The quadratic formula is a powerful tool for solving equations of the form ax^2 + bx + c = 0. It is given by:

q = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = 2, b = -85, and c = 100. Let's plug these values into the formula:

q = (-(-85) ± √((-85)^2 - 4 * 2 * 100)) / (2 * 2)

Now, let's simplify this expression step by step.

Applying the quadratic formula, q = (-b ± √(b^2 - 4ac)) / (2a), to our equation 2q^2 - 85q + 100 = 0 is a systematic process. Each step in the calculation is crucial for arriving at the correct solutions for q, which represent the output levels at which the total revenue equals 100. First, we substitute the coefficients a = 2, b = -85, and c = 100 into the formula, resulting in q = (-(-85) ± √((-85)^2 - 4 * 2 * 100)) / (2 * 2). The next step involves simplifying the expression inside the square root, which is the discriminant (b^2 - 4ac). The discriminant plays a crucial role in determining the nature of the roots – whether they are real or complex. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (a repeated root), and a negative discriminant indicates two complex roots. Calculating the discriminant in our case helps us understand the feasibility of achieving a total revenue of 100. The ± symbol in the quadratic formula indicates that there are potentially two solutions for q. This is because the square root of a positive number has two values, one positive and one negative. These two solutions represent the two points where the parabola intersects the horizontal line representing the total revenue of 100. Therefore, by carefully following the steps of the quadratic formula and simplifying the expression, we can determine the output levels that correspond to a total revenue of 100.

Simplifying the Quadratic Formula

Let's simplify the expression inside the square root first:

(-85)^2 - 4 * 2 * 100 = 7225 - 800 = 6425

Now, substitute this back into the equation:

q = (85 ± √6425) / 4

Next, we find the square root of 6425, which is approximately 80.16:

q = (85 ± 80.16) / 4

Now, we have two possible solutions for q, one with addition and one with subtraction.

Simplifying the quadratic formula involves a series of arithmetic operations, each crucial for arriving at the final solutions for q. The initial simplification focuses on the discriminant (b^2 - 4ac), which provides valuable information about the nature of the roots. In our case, calculating (-85)^2 - 4 * 2 * 100 results in 6425, a positive value. This positive discriminant confirms that the quadratic equation has two distinct real roots, meaning there are two output levels at which the total revenue equals 100. Substituting the discriminant back into the quadratic formula, we get q = (85 ± √6425) / 4. The next step involves finding the square root of 6425, which is approximately 80.16. This approximation is important for practical applications, as it provides a numerical value that can be used for decision-making. The ± symbol in the equation indicates that we need to consider both the positive and negative square roots, leading to two possible solutions for q. These two solutions represent the two points where the parabola representing the total revenue function intersects the horizontal line representing the target revenue of 100. Therefore, by carefully simplifying the expression and considering both the positive and negative square roots, we can determine the two output levels that satisfy the condition of a total revenue of 100.

Calculating the Two Possible Solutions

Let's calculate the two solutions for q:

Solution 1 (using addition): q1 = (85 + 80.16) / 4 = 165.16 / 4 ≈ 41.29

Solution 2 (using subtraction): q2 = (85 - 80.16) / 4 = 4.84 / 4 ≈ 1.21

So, we have two possible output levels: approximately 41.29 units and approximately 1.21 units.

Calculating the two possible solutions for q involves separating the ± symbol in the quadratic formula and performing the arithmetic operations for both the addition and subtraction cases. For the first solution, q1, we use the addition operation: q1 = (85 + 80.16) / 4. This results in 165.16 / 4, which is approximately 41.29. For the second solution, q2, we use the subtraction operation: q2 = (85 - 80.16) / 4. This results in 4.84 / 4, which is approximately 1.21. These two solutions, 41.29 units and 1.21 units, represent the output levels at which the total revenue is equal to 100. In the context of a business, these values provide valuable insights into the production levels that would generate the desired revenue. It's important to note that both solutions are valid mathematically, but in a practical business context, it's essential to consider other factors such as production capacity, market demand, and cost considerations to determine which output level is more feasible and profitable. For example, producing 41.29 units might require significant investment in additional resources, while producing 1.21 units might not be sufficient to cover the fixed costs. Therefore, the two solutions provide a range of options, and the optimal choice depends on a comprehensive analysis of the business environment.

Interpreting the Results

We found two output levels where the total revenue is 100: approximately 41.29 units and approximately 1.21 units. This means that if the business produces and sells either 1.21 units or 41.29 units, it will generate a total revenue of 100. The two solutions arise because the revenue function is a parabola. The revenue increases as the quantity increases up to a certain point (the vertex of the parabola) and then decreases. So, there are two points on the parabola where the revenue is 100.

Interpreting the results of the quadratic equation in the context of a business revenue function requires a careful consideration of the two solutions obtained. The fact that we found two output levels, approximately 41.29 units and approximately 1.21 units, where the total revenue is 100, is a direct consequence of the parabolic nature of the revenue function. The revenue function, TR = 85q - 2q^2, represents a parabola that opens downwards, indicating that there is a maximum revenue point. This means that as the quantity produced and sold increases from zero, the total revenue initially increases, reaches a peak, and then starts to decline. The two solutions we found represent the two points on the parabola where the revenue curve intersects the horizontal line representing a total revenue of 100. One solution (1.21 units) corresponds to a point on the ascending part of the parabola, where the revenue is increasing as quantity increases. The other solution (41.29 units) corresponds to a point on the descending part of the parabola, where the revenue is decreasing as quantity increases. This understanding is crucial for businesses because it highlights the concept of diminishing returns. Producing beyond the quantity that maximizes revenue can lead to a decrease in total revenue, even though more units are being sold. Therefore, while both solutions technically achieve a total revenue of 100, the business needs to carefully evaluate the implications of operating at each output level in terms of profitability and overall business strategy.

Practical Considerations

In a real-world business scenario, several factors need to be considered when interpreting these results. First, the business needs to assess its production capacity. Can it realistically produce 41.29 units? If its capacity is lower, then the solution of 1.21 units might be more relevant. Second, the business needs to consider its costs. Producing 41.29 units might require more resources and incur higher costs than producing 1.21 units. The business needs to ensure that the revenue generated covers these costs. Third, the business needs to analyze the market demand. Is there enough demand for 41.29 units? If not, producing that many units might lead to unsold inventory.

In addition to the mathematical solutions, practical considerations play a vital role in making informed business decisions. While the quadratic equation provides two output levels (41.29 units and 1.21 units) that yield a total revenue of 100, a business must assess the feasibility and implications of operating at each level. Production capacity is a crucial factor. A business must evaluate whether it has the resources, equipment, and workforce to produce 41.29 units. If its production capacity is limited, the solution of 1.21 units might be more realistic, even though it might not be the most profitable option. Cost considerations are also paramount. Producing 41.29 units might require significant investment in raw materials, labor, and overhead costs. The business must analyze whether the revenue generated from selling these units will sufficiently cover these costs and provide an acceptable profit margin. Producing 1.21 units, on the other hand, might have lower costs but might also result in lower overall profits. Market demand is another critical factor to consider. Even if a business has the capacity to produce 41.29 units, it needs to assess whether there is sufficient demand for that quantity in the market. Producing more units than can be sold can lead to unsold inventory, storage costs, and potential losses due to obsolescence or spoilage. Therefore, a business should not solely rely on the mathematical solutions but should also consider these practical factors to make a well-informed decision about the optimal output level.

Conclusion

In conclusion, we successfully determined the output levels at which the total revenue is 100 for the given function TR = 85q - 2q^2. By setting up the equation, rearranging it into standard quadratic form, and using the quadratic formula, we found two solutions: approximately 41.29 units and approximately 1.21 units. However, in a real-world business scenario, it's crucial to consider practical factors like production capacity, costs, and market demand to make the best decision. Understanding these concepts is vital for businesses aiming to optimize their revenue and profitability. For further insights into revenue optimization, explore resources available on trusted websites such as Investopedia's Revenue Definition.