Problem 46 - Algebra A Combined Function

Question

Recall that in business, a demand function expresses the quantity of a commodity demanded as a function of the commodity's unit price. A supply function expresses the quantity of a commodity supplied as a function of the commodity's unit price. When the quantity produced and supplied is equal to the quantity demanded, then we have what is called market equilibrium. Use this information for Exercises 45 and \(46 .\) The demand function for a certain style of picture frame is given by the function \(p(x)=-2 x^{2}+90\) and the corresponding supply function is given by \(p(x)=9 x+34,\) where \(p(x)\) is in dollars and \(x\) is in thousands of units. Find the equilibrium quantity and the corresponding price by solving the system consisting of the two given equations.

Step-by-Step Solution

Answer

Equilibrium quantity: 3.25 units; Equilibrium price: $63.25.

Step 1: Set Demand Equal to Supply

To find the market equilibrium, set the demand function equal to the supply function. This means solving the equation: \[-2x^2 + 90 = 9x + 34\]

Step 2: Rearrange the Equation

Move all terms to one side of the equation to set it to 0:\[-2x^2 + 90 - 9x - 34 = 0\]Simplify this to:\[-2x^2 - 9x + 56 = 0\]

Step 3: Solve the Quadratic Equation

Use the quadratic formula to solve \(-2x^2 - 9x + 56 = 0\). The quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a = -2\), \(b = -9\), and \(c = 56\).

Step 4: Compute Discriminant

Calculate the discriminant \(b^2 - 4ac = (-9)^2 - 4(-2)(56)\).This simplifies to:\[81 + 448 = 529\]

Step 5: Apply the Quadratic Formula

Substitute into the quadratic formula:\[x = \frac{9 \pm \sqrt{529}}{-4}\]Compute the square root and solve:\[x = \frac{9 \pm 23}{-4}\]This gives two solutions, \(x_1 = -8\) and \(x_2 = 3.25\).

Step 6: Select the Feasible Solution

Since the quantity \(x\) must be non-negative, the feasible solution is \(x = 3.25\).

Step 7: Determine the Equilibrium Price

Substitute \(x = 3.25\) back into either the demand or supply function to find the equilibrium price. Using the supply function:\[p(3.25) = 9(3.25) + 34 = 29.25 + 34 = 63.25\]The equilibrium price is \(\$63.25\).

Key Concepts

Demand Function

The demand function is an essential concept in economics, representing how quantity demanded changes with price. In this exercise, the demand function is given by the formula \(p(x) = -2x^2 + 90\). This equation showcases a quadratic relationship, meaning as the price varies, the quantity demanded responds non-linearly. Here's how it works:

• The coefficient \(-2x^2\) implies that as more units are demanded (or as \(x\) increases), the price effect decreases quadratically.
• The constant 90 is the starting price when no items are demanded.
• Demand generally decreases as price increases, which is what this downward-opening quadratic equation typically represents.

Understanding the demand function's shape and direction helps anticipate how price changes affect market dynamics.

Supply Function

The supply function depicts the relationship between the price of a commodity and the amount a supplier is willing to produce and sell. For this exercise, the supply function is expressed as \(p(x) = 9x + 34\). This is a linear function, indicating a consistent change in price with each additional thousand units supplied.

• The coefficient \(9x\) represents the increase in price per additional unit supplied. Here, more units result in higher prices.
• The constant 34 is the initial price when no units are supplied.
• Unlike the demand function, which decreases with increasing quantity, the supply function increases, reflecting a standard economic principle where higher production levels correlate with higher prices.

This linear relationship makes it easier to predict how changes in supply affect the price, particularly useful for determining market strategies.

Quadratic Equation

The quadratic equation is pivotal in solving both the demand and supply functions to find the equilibrium. In this exercise, we derive the equation \(-2x^2 - 9x + 56 = 0\) after setting the demand function equal to the supply function. Here's how we solve a quadratic equation:

• Identify coefficients: In our equation, \(a = -2\), \(b = -9\), and \(c = 56\).
• Compute the discriminant using \(b^2 - 4ac\). A discriminant of 529 indicates two real and distinct solutions.
• Solve the quadratic equation using the formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

By plugging in the values, we find two potential solutions for \(x\). Understanding quadratic equations is crucial as it reveals possible quantities where market demand and supply balance.

Equilibrium Price

Finding the equilibrium price is the ultimate goal of combining demand and supply functions. It represents the price at which the quantity demanded equals the quantity supplied, indicating market stability. From our solution, after solving the quadratic equation, we find the feasible quantity is \(x = 3.25\). By substituting this back into the supply function, we calculate the equilibrium price:

• Substitute \(x = 3.25\) into the supply equation: \(p(x) = 9(3.25) + 34\).
• Complete the calculation: \(p(3.25) = 29.25 + 34 = 63.25\).

Thus, the equilibrium price in this scenario is \(\$63.25\). Understanding how to determine the equilibrium price helps predict how markets react to changes in supply and demand, guiding pricing and production decisions.