Problem 45 - 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 compact disc is given by the function \(p(x)=-0.01 x^{2}-0.2 x+9\) and the corresponding supply function is given by \(p(x)=0.01 x^{2}-0.1 x+3,\) 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 is 20, and the price is \( \$1.00 \).

Step 1: Understand the Problem

We need to find the equilibrium quantity and price. This occurs when the demand and supply functions are equal. Given functions are the demand function \( p(x) = -0.01x^2 - 0.2x + 9 \) and the supply function \( p(x) = 0.01x^2 - 0.1x + 3 \). We need to solve the equation \( -0.01x^2 - 0.2x + 9 = 0.01x^2 - 0.1x + 3 \).

Step 2: Set Demand Equal to Supply

Equate the demand and supply functions: \( -0.01x^2 - 0.2x + 9 = 0.01x^2 - 0.1x + 3 \). This equation will allow us to find the value of \( x \), which is the equilibrium quantity in thousands of units.

Step 3: Simplify the Equation

Rearrange and combine like terms: \( -0.01x^2 - 0.2x + 9 = 0.01x^2 - 0.1x + 3 \) becomes \(-0.02x^2 - 0.1x + 6 = 0 \).

Step 4: Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -0.02 \), \( b = -0.1 \), and \( c = 6 \). Calculate \( b^2 - 4ac = (-0.1)^2 - 4(-0.02)(6) = 0.01 + 0.48 = 0.49 \). Thus, \( x = \frac{0.1 \pm \sqrt{0.49}}{-0.04} \).

Step 5: Calculate the Solutions for x

Calculate \( \sqrt{0.49} = 0.7 \). Substitute back into the formula to get: \( x = \frac{0.1 + 0.7}{-0.04} = 20 \) or \( x = \frac{0.1 - 0.7}{-0.04} = -15 \). The negative solution is not feasible, so \( x = 20 \).

Step 6: Find the Equilibrium Price

Substitute \( x = 20 \) back into either the demand or supply function to find \( p(x) \). Using the demand function: \( p(20) = -0.01(20)^2 - 0.2(20) + 9 \). Calculate: \( p(20) = -4 - 4 + 9 = 1 \). The equilibrium price is \( \$1.00 \).

Key Concepts

Demand Function

A demand function represents the relationship between the price of a commodity and the quantity demanded by consumers. In our exercise, the demand function is given by \( p(x) = -0.01x^2 - 0.2x + 9 \). Here, \( p(x) \) is the price in dollars, while \( x \) is the quantity in thousands of units.
The demand function often has a downward slope. This is because as the price decreases, more consumers are willing to purchase the product, thus increasing demand. Conversely, as the price goes up, the quantity demanded generally decreases because fewer consumers are willing or able to buy.

• Negative coefficients, like \(-0.01x^2\) and \(-0.2x\), show the inverse relationship between price and quantity demanded.

Understanding this relationship helps businesses and policymakers make informed decisions about pricing strategies and market interventions.

Supply Function

A supply function illustrates how much of a commodity producers are willing to supply at different price levels. Our exercise gives us the supply function as \( p(x) = 0.01x^2 - 0.1x + 3 \). As before, \( p(x) \) indicates the price in dollars and \( x \) is the amount supplied in thousands of units.
Unlike the demand function, the supply function typically has a positive slope. This indicates that as prices rise, producers are more inclined to sell more of the good because they get higher returns, and vice versa when prices fall.

• The positive \(0.01x^2\) term indicates the direct relationship between price and quantity supplied.

This relationship is crucial for companies to understand when deciding how much of a product to bring to market. It ensures they can meet demand without overproduction, which could result in excess inventory.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation that often appears in the standard form \( ax^2 + bx + c = 0 \). Solving these equations allows us to find the values of \( x \) that make the equation true.
In our exercise, we set the demand and supply equations equal, which simplifies to \(-0.02x^2 - 0.1x + 6 = 0\) after rearranging and combining like terms. This is a quadratic equation, where:

• \(a = -0.02\)
• \(b = -0.1\)
• \(c = 6\)

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), helps us solve for \(x\). For the exercise, this results in the solutions \(x = 20\) and \(x = -15\). Since negative quantities don't make sense in this context, \(x = 20\) is chosen. This demonstrates the method for finding the equilibrium quantity where supply meets demand.

Equilibrium Price

The equilibrium price is reached when the quantity of a good demanded by consumers equals the quantity supplied by producers. In our context, once we determined \(x = 20\) as the equilibrium quantity (in thousands), we needed to find the corresponding equilibrium price.
Substituting \(x = 20\) back into either the demand or supply function, we calculated \(p(20)\) using the demand function: \(p(20) = -0.01(20)^2 - 0.2(20) + 9\). The calculations yielded \(p(20) = 1\), meaning the equilibrium price is \$1.00.
This price is where the market is perfectly balanced, preventing either surplus (too much supply) or shortage (too much demand). Understanding equilibrium helps businesses avoid lost sales or wasted resources.