Problem 75 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the point of equilibrium of the demand and supply equations. The point of equilibrium is the price \(p\) and the number of units \(x\) that satisfy both the demand and supply equations. Demand \(\quad\) Supply \(p=140-0.00002 x \quad p=80+0.00001 x\)

Step-by-Step Solution

Answer

The point of equilibrium for the supply and demand equations is about \(x = 2,000,000\) units and the price \(p = 100\).

Step 1: Setting the Equations Equal to Each Other

First, identify the point of equilibrium by setting the demand and supply equations equal to each other. The point of equilibrium can be found as follows: \n\(140-0.00002x = 80+0.00001x\)

Step 2: Solve for x

Next, arrange the equation to solve for x. Combine like terms and simplify. The equation will become:\n\(0.00003x = 60\). Now, solve for x by dividing both sides of the equation by 0.00003, thus, \n\(x = 60 / 0.00003\).

Step 3: Substitute x into one of the Original Equations

Now that we have the value for x, substitute it into one of the original equations, either the demand or supply equation, to find the corresponding price (p). For instance, substituting x into the demand equation, will be: \n\(p = 140 - 0.00002x\)

Key Concepts

Demand Equation

A demand equation represents the relationship between the price of a product and the quantity demanded by consumers. It typically shows that as the price of an item decreases, more people are willing to purchase it, hence the negative relationship between price and quantity. In our problem, the demand equation is given as:

• \( p = 140 - 0.00002x \)

Here, \( p \) is the price of the product, and \( x \) is the quantity demanded. The equation implies that with each increment of one unit in demand, the price decreases slightly by \( 0.00002 \). This negative slope is typical for demand equations as they illustrate the inverse relationship between price and demand in a market where decreasing prices lead to higher consumer purchases. Understanding how to interpret these equations is crucial for determining how quantity changes with varying prices.

Supply Equation

A supply equation, on the other hand, shows the relationship between the price of a product and the amount of the product that suppliers are willing to provide. Generally, suppliers are more inclined to increase the quantity offered when prices are higher, indicating a direct relationship between price and quantity supplied. In this exercise, the supply equation is defined as:

• \( p = 80 + 0.00001x \)

Again, \( p \) is the price, while \( x \) symbolizes the quantity supplied. As we can see, this time the coefficient of \( x \) is positive (\( 0.00001 \)), which typically characterizes supply equations. Here, for each additional unit supplied, the price slightly increases, embodying the positive slope that indicates a direct correlation between price and supply. Analyzing these equations helps in understanding how suppliers adjust their offerings in response to price changes.

Solving Equations

Finding the equilibrium point requires solving equations - a fundamental mathematical skill. The equilibrium occurs where the quantity demanded by consumers equals the quantity supplied by producers. This can be achieved by setting the demand and supply equations equal and solving for the number of units (\( x \)) and the price (\( p \)).To solve, we equate the two equations:

• \( 140 - 0.00002x = 80 + 0.00001x \)

By simplifying this, we combine like terms:

• \( 0.00003x = 60 \)

Then, solving for \( x \), we divide both sides by \( 0.00003 \):

• \( x = \frac{60}{0.00003} \)

After finding \( x \), we substitute it back into either the demand or supply equation to determine \( p \). By understanding the process of solving these equations, one gains insight into determining equilibrium - a vital concept in economics that indicates a stable market state where supply meets demand.