Problem 50 - Intermediate Algebra

Question

The demand function for a certain style of picture frame is given by the function $$ p=-2 x^{2}+90 $$ and the corresponding supply function is given by $$ p=9 x+34 $$ where \(p\) 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 is 3.5 thousand units, and the price is $65.5.

Step 1: Understand the Problem

We need to find the equilibrium quantity and price where the demand and supply are equal, meaning the two equations for demand and supply must intersect.

Step 2: Set Equations Equal to Each Other

Set the demand equation and the supply equation equal to solve for the equilibrium quantity, \(-2x^2 + 90 = 9x + 34\).

Step 3: Rearrange into a Standard Quadratic Equation

Rearrange the equation \(-2x^2 + 90 = 9x + 34\) into a standard form by bringing all terms to one side: \(-2x^2 - 9x + 90 - 34 = 0\). Simplify to get \(-2x^2 - 9x + 56 = 0\).

Step 4: Simplify the Equation

If possible, simplify the quadratic equation by factoring or use the quadratic formula. Our equation \(-2x^2 - 9x + 56 = 0\) is already in its simplest form, so we'll use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 5: Apply the Quadratic Formula

Identify \(a = -2\), \(b = -9\), and \(c = 56\) from the equation. Substitute these values into the quadratic formula: \(x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(-2)(56)}}{2(-2)}\).

Step 6: Calculate Discriminant

Calculate the discriminant \(b^2 - 4ac = (-9)^2 - 4(-2)(56) = 81 + 448 = 529\). Take the square root of the discriminant: \(\sqrt{529} = 23\).

Step 7: Find Values of x

Substitute back into the quadratic formula: \(x = \frac{9 \pm 23}{-4}\) to find the potential values for \(x\). Calculate for each case.

Step 8: Validate Solutions

Calculate \(x_1= \frac{9 + 23}{-4} = -8\) (not valid since quantity cannot be negative) and \(x_2 = \frac{9 - 23}{-4} = 3.5\). Only \(x = 3.5\) is valid.

Step 9: Find Equilibrium Price

Substitute \(x = 3.5\) into one of the original equations, for instance the supply equation: \(p = 9(3.5) + 34\). Calculate \(p = 31.5 + 34 = 65.5\).

Key Concepts

Demand and Supply Equations

In economics, the demand and supply equations are pivotal for understanding how markets operate. These equations represent two fundamental economic concepts: the amount of a product that consumers are willing to buy at various prices (demand) and the amount that producers are willing to sell (supply).
The demand equation is typically a downward sloping curve, which in our exercise is represented as \( p = -2x^2 + 90 \). Here, \( p \) is the price and \( x \) is the quantity in thousands. It tells us that as the quantity increases, the price decreases due to less demand.
The supply equation, on the other hand, is usually an upward sloping curve, represented by \( p = 9x + 34 \) in this problem. It shows that as more quantity is supplied, the price tends to increase as a result of higher production costs.
To find the equilibrium where market supply matches demand, we set these two equations equal. This intersection point gives us the equilibrium quantity and price, which is the focus of many economic studies.

Quadratic Formula

The quadratic formula is a powerful mathematical tool used to find the roots of a quadratic equation of the form \( ax^2 + bx + c = 0 \). It's expressed as:

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

This formula is derived from completing the square method and provides a solution for any quadratic equation where the discriminant \( b^2 - 4ac \) is non-negative.
In our exercise, after rearranging our demand and supply equations into the standard quadratic form \( -2x^2 - 9x + 56 = 0 \), we identify \( a = -2 \), \( b = -9 \), and \( c = 56 \).
By inserting these values into the quadratic formula, we can solve for \( x \). Calculating the discriminant, \( 529 \), confirms the presence of real roots. This formula helps us find \( x_1 = -8 \) and \( x_2 = 3.5 \), but only \( x = 3.5 \) is valid since quantities cannot be negative.

Solving Systems of Equations

Solving systems of equations involves finding the set of values that satisfy all equations simultaneously. When dealing with the demand and supply equations, we're essentially solving a system of equations to find where they intersect.
Firstly, we align the demand and supply equations: \(-2x^2 + 90 = 9x + 34 \). This allows us to find the values of \( x \), which represent the equilibrium quantity.
The system is solved by setting the equations equal and rearranging them into a single quadratic equation: \( -2x^2 - 9x + 56 = 0 \). This step converts the problem into a format where the quadratic formula can be applied, showing the intersection points.
Finally, once the quantity is found, it's crucial to substitute back into either the supply or demand equation to determine the equilibrium price. This verification step ensures we've accurately located the equilibrium in price terms. The systematic approach verifies the consistency of the solution and its utility in economic models.