Problem 65 - Algebra and Trigonometry

Question

Revenue A manufacturer finds that the revenue generated by selling \(x\) units of a certain commodity is given by the function \(R(x)=80 x-0.4 x^{2}\) , where the revenue \(R(x)\) is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum?

Step-by-Step Solution

Answer

The maximum revenue is $4000 from manufacturing 100 units.

Step 1: Identify the Function

The function given is a quadratic function: \[ R(x) = 80x - 0.4x^2 \]This function represents the revenue generated by selling \(x\) units of a commodity.

Step 2: Find the Maximum Point

To find the maximum revenue, we need to identify the vertex of the parabola represented by the quadratic equation. For a parabola \( ax^2 + bx + c \), the vertex occurs at \( x = -\frac{b}{2a} \).In our equation, - \( a = -0.4 \)- \( b = 80 \)Thus, the vertex \(x\)-value is:\[ x = -\frac{80}{2(-0.4)} = 100 \]

Step 3: Substitute Back to Find Maximum Revenue

Substitute \( x = 100 \) back into the revenue function to find the maximum revenue:\[R(100) = 80(100) - 0.4(100)^2 = 8000 - 4000 = 4000\]So, the maximum revenue is \$4000.

Key Concepts

Maximum Revenue

In business, maximizing revenue is essential for profitability and sustainability. Understanding how to calculate the maximum revenue using a quadratic revenue function is crucial. The function given in the exercise, \[ R(x) = 80x - 0.4x^2 \] represents a revenue model where revenue depends on the number of units, \( x \), being produced and sold. The goal is to determine the maximum value that this revenue function can reach, which is known as the maximum revenue. The method for finding this maximum involves calculus and algebra, specifically through finding the vertex of the parabola formed by the quadratic function. Once the vertex is found, the maximum revenue corresponds to the \( R(x) \) value at this point. Identifying this maximum point is beneficial not only in theoretical exercises but also in practical business scenarios where optimal production levels are needed.

Parabola Vertex

The vertex of a parabola is a point that signifies the maximum or minimum of that quadratic function, depending on whether the parabola opens upwards or downwards. In the context of the equation:\[ R(x) = 80x - 0.4x^2 \]we're dealing with a parabola that opens downwards since the coefficient of the \( x^2 \) term is negative.To find the vertex, use the standard formula for the vertex of a quadratic equation, \( ax^2 + bx + c \), which is at \( x = -\frac{b}{2a} \).For our equation:

• \( a = -0.4 \)
• \( b = 80 \)

Plug these into the formula:\[ x = -\frac{80}{2(-0.4)} = 100 \]The vertex is at \( x = 100 \), which tells us the number of units where the revenue is maximized. Thus, the vertex directly provides the production level for maximum revenue.

Revenue Function

The revenue function in our problem,\[ R(x) = 80x - 0.4x^2 \],is a quadratic function that relates revenue \( R \) to the number of units \( x \). Quadratic functions like these are crucial in economics and business for modeling how varying quantities affect revenue.In this function:

• \( 80x \) represents the revenue per unit times the number of units.
• \(-0.4x^2 \) depicts diminishing returns, showing how increased production eventually leads to lower additional revenue.

This type of function is typical in scenarios where there's an initial increase in profit as production rises, followed by a decrease due to factors like market saturation or increased cost per unit as output increases. Recognizing these elements helps businesses analyze economic conditions and make informed decisions on production levels.

Units of Production

Understanding the role that units of production play in a revenue function is essential for achieving maximum revenue. In our exercise, you determine the optimal number of units by finding the vertex of the parabola described by the revenue function.Units of production are represented by \( x \) in \[ R(x) = 80x - 0.4x^2 \].The calculated vertex is at \( x = 100 \), meaning that to achieve maximum revenue, 100 units should be manufactured. Producing more or less than this optimum number can result in decreased revenue due to inefficiencies or market saturation. Optimizing units of production ensures that the company operates within its most financially beneficial capacity, balancing the costs and returns effectively. Thus, understanding and manipulating the number of units produced, based on the revenue function, is a key strategy in maximizing economic success.