Problem 65 - Algebra and Trigonometry

Question

The table shows the price of a gallon of unleaded premium gasoline. For each price, the table lists the number of gallons per day that a gas station sells and the number of gallons per day that can be supplied. $$\begin{array}{lll}{\text { Price per }} & {\text { Gallons Demanded }} & {\text { Gallons Supplied }} \\ {\text { Gallon }} & {\text { per Day }} & {\text { per Day }} \\ {\$ 3.20} & {1400} & {200} \\ {\$ 3.60} & {1200} & {600} \\ {\$ 4.40} & {800} & {1400} \\ {\$ 4.80} & {600} & {1800}\end{array}$$ The data in the table are described by the following demand and supply models: Demand Model \(\quad\) Supply Model \(p=-0.002 x+6 \quad p=0.001 x+3\) a. Solve the system and find the equilibrium quantity and the equilibrium price for a gallon of unleaded premium gasoline. b. Use your answer from part (a) to complete this statement: If unleaded premium gasoline is sold for _____ per gallon, there will be a demand for ______ gallons per day and ______ gallons will be supplied per day.

Step-by-Step Solution

Answer

a. The equilibrium quantity and price for a gallon of unleaded premium gasoline are 1000 gallons and $4, respectively. b. If unleaded premium gasoline is sold for $4 per gallon, there will be a demand for 1000 gallons per day, and 1000 gallons will be supplied per day.

Step 1: Set up the system of equations

Create a system of equations using the given demand and supply models. This results in the following system of equations: \n-0.002x + 6 = 0.001x + 3.

Step 2: Solve the system of equations

Solve the system of equations initialized in Step 1. Add 0.002x to each side to eliminate the term on the left side and subtract 3 from each side to eliminate the term on the right side. This simplifies to: 0.003x = 3. Then divide each side by 0.003 to solve for x. The solution is x = 1000.

Step 3: Find the equilibrium price

Now that the equilibrium quantity is known, it can be used to find the equilibrium price. Substitute x = 1000 back into either the demand or supply equation to solve for p. Substituting x = 1000 into the supply equation results in p = 0.001*1000 + 3 = 4. Hence, when 1000 gallons per day are bought and sold, the equilibrium price of a gallon of unleaded premium gasoline is $4.

Step 4: Complete the statement in part (b)

Using the results from the previous steps, fill in the blanks in the statement from part b. If unleaded premium gasoline is sold for $4 per gallon, there will be a demand for 1000 gallons per day, and 1000 gallons will be supplied per day.

Key Concepts

Demand and Supply Models

Demand and supply models are vital tools used to illustrate the relationship between the quantity of a good that consumers are willing to buy and the quantity that producers are willing to supply at different prices. These models employ equations to represent how changing the price of a good affects its demand and supply. For example, a demand model might be expressed in an equation like \[ p = -0.002x + 6 \]where

• p is the price per unit,
• x is the quantity demanded.

Similarly, a supply model could be \[ p = 0.001x + 3 \]In this context, when demand decreases as the price rises and supply increases with a price increase, these linear equations provide a useful approximation of market behavior. By using these models, we can predict how different prices will affect the quantity bought and sold.

System of Linear Equations

When analyzing market equilibrium, economists use a system of linear equations to find the point where demand equals supply. This involves solving two equations simultaneously. In our gasoline example, we solve the system:\[\begin{align*}-0.002x + 6 &= 0.001x + 3\end{align*}\]This equation represents both the demand and supply models intersecting. To solve the system, we isolate and solve for the variable \( x \), which represents the quantity in this context. By manipulating the equation:

• Add \(0.002x\) to both sides to remove it from the left side.
• Subtract 3 from both sides, simplifying the equation to \( 0.003x = 3 \).
• Divide both sides by \(0.003\) to solve for \(x\).

This system tells us the quantity at which supply matches demand, an essential computation for price setting. Interpreting these results is crucial for understanding how equilibrium is found in markets.

Equilibrium Price

The equilibrium price is a central concept in economics that represents the price at which the quantity of goods supplied matches the quantity of goods demanded. Using the equilibrium quantity derived from our system of equations, we can find the equilibrium price by substituting back into either the supply or demand equation. In our example of gasoline:Using the supply equation \[ p = 0.001 imes 1000 + 3 \]we find \[ p = 4 \]This means when 1000 gallons are demanded and supplied per day, the market price settles at $4 per gallon. This equilibrium ensures that the market clears, meaning there is neither a surplus nor a shortage. Understanding this concept helps businesses and policymakers anticipate and respond to market dynamics effectively.