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Chapter 2: Problem 27

For the supply equations in Exercises 27 and 28, where \(x\) is the quantity supplied in units of a thousand and \(p\) is the unit price in dollars, (a) sketch the supply curve and (b) determine the price at which the supplier will make 2000 units of the commodity available in the market. \(p=2 x^{2}+18\)

Short Answer

Expert verified
In brief, the given supply equation is \(p = 2x^2 + 18\), with \(x\) in thousands of units. To sketch the supply curve, we find that the vertex of the parabola is at (0, 18), indicating the minimum price when no units are supplied. The curve is an upward-opening parabola. To find the price at which the supplier will supply 2000 units, we substitute \(x = 2\), corresponding to 2000 units, into the supply equation: \(p = 2(2^2) + 18 = 26\). Thus, the required price is $26 per unit.

Step by step solution

01

Plot the Supply Curve

To plot the supply curve, we need to graph the equation \(p = 2x^2 + 18\). By recognizing this is a parabolic equation with a positive quadratic term, we can tell that it is a parabola opening upwards. To help sketch the curve, we can find the vertex: This will be where the price is at its minimum value. To find the vertex, we can use the formula \(x_v = -\frac{b}{2a}\), where \(a = 2\) and \(b = 0\) for our equation. Thus, \[x_v = -\frac{0}{2(2)} = 0\] Substituting the value back into the equation, we find the price at the vertex: \[p_v = 2(0)^2 + 18 = 18\] So, the vertex is at the point (0, 18), which is when no units are supplied. Now we can sketch the curve using this information. The plot will show an upward-opening parabola with a minimum at (0, 18).
02

Find the Price at the Desired Quantity

We are given that the supplier will make 2000 units of the commodity available. Since the quantity supplied is given in thousands, we will divide by 1000 to get the value of \(x\): \[x = \frac{2000}{1000} = 2\] Now, substitute \(x=2\) into the supply equation to find the corresponding price \(p\): \[p = 2(2^2) + 18 = 2(4) + 18 = 8 + 18 = 26\] Thus, the price at which the supplier will supply 2000 units of the commodity is $26 per unit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Parabolic Equations
Parabolic equations describe the shape of a parabola, a curve that is often used to model various phenomena in physics, biology, and economics. In our exercise, the supply curve equation represents a parabolic curve and is written as p = 2x^2 + 18, where p represents the price per unit and x the quantity in thousands.

Because the coefficient in front of the x^2 term is positive, the parabola opens upwards. This particular shape suggests that as the quantity increases, the price rises at an increasing rate, which is typical in many supply scenarios where higher production quantities lead to increased costs. Understanding this graphical representation is crucial for economic analysis. When sketching the parabola, the vertex serves as the starting point, and since our equation lacks a x term, the vertex is at the origin (0, 18), indicating that no units are supplied at the base price.
Quadratic Functions and Their Features
Quadratic functions, such as p = 2x^2 + 18 from our problem, are second-degree polynomials and have several key features that are important for solving and graphing the equation.

The general form of a quadratic function is y = ax^2 + bx + c. It is known for creating a U-shaped curve called a parabola. The a coefficient determines whether the parabola opens up or down, while the b and c values impact the location of the parabola's vertex and axis of symmetry. For the given supply function, the vertex's price (p_v) was found using the coordinates of the vertex and is considered the minimum price in this context because our parabola opens upwards.
Economics Mathematics: Analyzing Supply Curves
In economics mathematics, the supply curve represents the relationship between the price of a good and the quantity of the good that a producer is willing to supply. It is an essential tool for understanding market dynamics.

In our exercise, we are given the equation p = 2x^2 + 18 to determine the supply curve. By calculating the price at which the supplier is willing to make 2000 units available, we perform a fundamental economic calculation: determining the price point for a specific supply level. The calculation involved substituting the quantity x into the supply equation to find p, revealing that the supplier will supply 2000 units at a price of $26 per unit. This critical point on the supply curve can inform decisions for suppliers and buyers in the marketplace.

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