Problem 49 - Intermediate Algebra

Question

The demand function for a certain compact disc is given by the function $$ p=-0.01 x^{2}-0.2 x+9 $$ and the corresponding supply function is given by $$ p=0.01 x^{2}-0.1 x+3 $$ 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

The equilibrium quantity is 20,000 units, and the price is $1.

Step 1: Set Up the Equilibrium Condition

The equilibrium point occurs where the demand and supply curves intersect. This means we set the demand function equal to the supply function:\[-0.01x^2 - 0.2x + 9 = 0.01x^2 - 0.1x + 3\]

Step 2: Simplify the Equation

Move all terms to one side of the equation to set it to zero:\[-0.01x^2 - 0.2x + 9 - 0.01x^2 + 0.1x - 3 = 0\]Combine like terms:\[-0.02x^2 - 0.1x + 6 = 0\]

Step 3: Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for \(x\), where \(a = -0.02\), \(b = -0.1\), and \(c = 6\): First, calculate the discriminant:\[(-0.1)^2 - 4(-0.02)(6) = 0.01 + 0.48 = 0.49\]Then calculate the roots:\[x = \frac{-(-0.1) \pm \sqrt{0.49}}{2(-0.02)}\]\[x = \frac{0.1 \pm 0.7}{-0.04}\]This gives two solutions for \(x\):\[x_1 = -20, \quad x_2 = 20\]

Step 4: Interpret the Physical Meaning

Since \(x\) represents the quantity of units sold in thousands, a negative quantity doesn't make sense in this context. Therefore, the feasible solution is \(x = 20\).

Step 5: Find the Equilibrium Price

Substitute \(x = 20\) back into one of the original equations to find \(p\). Let's use the demand equation:\[p = -0.01(20)^2 - 0.2(20) + 9\]\[p = -0.01(400) - 4 + 9\]\[p = -4 - 4 + 9 = 1\]Thus, the equilibrium price is \(1\) dollar.

Key Concepts

Demand Function

In economics, the demand function is a critical concept that describes the relationship between the price of a good and the quantity demanded by consumers. It's usually expressed in equation form, where the independent variable is the price, and the dependent variable is the quantity demanded.
For our example, the demand function is given by:

• \( p = -0.01x^2 - 0.2x + 9 \)

Here, \( p \) represents the price in dollars, and \( x \) is the quantity in thousands of units. The negative coefficients indicate that as the price rises, the quantity demanded decreases, which is typical for most demand functions. This downward slope reflects the law of demand: consumers buy less when prices go up and more when prices drop.
Understanding how to interpret this equation helps in predicting consumer behavior and making informed business decisions.

Supply Function

The supply function, much like the demand function, illustrates the relationship between the price of a good and the quantity that producers are willing to supply. It is generally an upward-sloping curve to reflect that higher prices encourage producers to supply more.
In our exercise, the supply function is expressed as:

• \( p = 0.01x^2 - 0.1x + 3 \)

Here, the positive \( x^2 \) term indicates that as the price increases, producers are more willing to increase the quantity supplied, creating an upward trend. This characteristic reflects the law of supply.
By analyzing the supply function, businesses can anticipate how much of a product will be available at different price points and plan their production accordingly.

Quadratic Equation

A quadratic equation is an algebraic expression that involves an unknown variable raised to the second power. It typically takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Solving these can help find the points of intersection for curves like supply and demand.
To find the intersection point—or equilibrium—of the supply and demand functions, we equate them and rearrange terms into a quadratic format:

• \( -0.02x^2 - 0.1x + 6 = 0 \)

The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is then used to solve for \( x \). This formula provides possible values for \( x \) where the curves intersect, balancing supply and demand.
Knowing how to solve quadratic equations is essential as it reveals vital economic points like equilibrium quantity.

Price

Price, an essential element in supply and demand analysis, determines the cost at which goods are sold and purchased in the market. In equilibrium, price is where the quantity supplied equals the quantity demanded.
In the exercise, after calculating the equilibrium quantity \( x = 20 \), we substitute this value into one of the original equations to find the equilibrium price:

• Using the demand function \( p = -0.01(20)^2 - 0.2(20) + 9 \)
• We find \( p = 1 \) dollar.

This price represents the point at which consumers and producers agree on the amount to be traded, balancing both sides. Evaluating prices helps economists and business leaders make strategic pricing and production decisions, aligning market strategies with consumer expectations.