Problem 112 - Elementary and Intermediate Algebra

Question

Supply and Demand. The number of mirrors that will be produced at a given price can be predicted by the formula \(s=\sqrt{23 x},\) where \(s\) is the supply (in thousands) and \(x\) is the price (in dollars). The demand \(d\) for mirrors can be predicted by the formula \(d=\sqrt{312-2 x^{2}} .\) Find the equilibrium price - that is, find the price at which supply will equal demand.

Step-by-Step Solution

Answer

The equilibrium price is $8.

Step 1: Understand the Formulas

We are given two formulas: The supply equation is \( s = \sqrt{23x} \) where \( s \) is in thousands and \( x \) is the price in dollars. The demand equation is \( d = \sqrt{312 - 2x^2} \). Our goal is to find the equilibrium price, which is when supply equals demand.

Step 2: Set Supply Equal to Demand

To find the equilibrium price, set the supply function equal to the demand function: \( \sqrt{23x} = \sqrt{312 - 2x^2} \). This equation states that when supply equals demand, both sides of the equation must be equal.

Step 3: Square Both Sides

To eliminate the square roots, square both sides of the equation: \((\sqrt{23x})^2 = (\sqrt{312 - 2x^2})^2\), which simplifies to \(23x = 312 - 2x^2\).

Step 4: Rearrange the Equation

Rearrange the equation to bring all terms to one side: \(2x^2 + 23x - 312 = 0\). This is a quadratic equation in standard form.

Step 5: Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = 23\), and \(c = -312\). Substitute these values into the formula to find \(x\).

Step 6: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\): \((23)^2 - 4 \cdot 2 \cdot (-312) = 529 + 2496 = 3025\). The discriminant is positive, indicating there are two real solutions.

Step 7: Apply the Quadratic Formula

Substitute the discriminant back into the quadratic formula: \(x = \frac{-23 \pm \sqrt{3025}}{4}\). Calculate \(\sqrt{3025}\) which is \(55\).

Step 8: Find the Roots

Calculate the roots: \(x = \frac{-23 + 55}{4} = 8 \) and \(x = \frac{-23 - 55}{4} = -19.5\). Since the price cannot be negative, only the positive solution \(x = 8\) is valid.

Step 9: Verify the Solution

Substitute \(x = 8\) back into the supply equation \(s = \sqrt{23 \times 8}\) and demand equation \(d = \sqrt{312 - 2 \times 8^2}\) to ensure they are equal, confirming the equilibrium price.

Key Concepts

Supply and Demand in Economics

In economics, the concept of supply and demand plays a crucial role in understanding how markets function. Supply refers to the amount of a product or service that producers are willing to offer at a given price, while demand represents the amount that consumers are willing to purchase. The relationship between supply and demand determines the market equilibrium. This is the point where the quantity supplied equals the quantity demanded.
To grasp this concept easily:

• Think of supply as the seller's willingness to sell products at various prices.
• Demand reflects the buyer's desire to purchase products based on their price.
• When supply equals demand, the market is in equilibrium, meaning there is no shortage or surplus.

Understanding this framework can help predict how any change in either supply or demand will affect pricing and availability in a real-world market scenario.

Understanding Quadratic Equations

Quadratic equations are an essential part of high school algebra, particularly when modeling real-world problems like calculating the equilibrium price in economics. A quadratic equation is typically represented in the form: \[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are coefficients, and \(x\) is the variable. In our supply and demand problem:

• We start with the supply and demand functions, which are set equal to each other to find the equilibrium price.
• This leads us to a scenario where we have a quadratic equation, here: \(2x^2 + 23x - 312 = 0\).
• Solving this requires using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
• By finding the values of \(x\), we determine the possible prices where supply and demand are equal.

Quadratic equations are powerful tools for analyzing situations where relationships are not linear, making them invaluable in diverse disciplines like physics, engineering, and economics.

The Role of Educational Algebra

Algebra serves as the foundational building block for more advanced mathematical concepts, including solving real-world problems in economics, like finding equilibrium prices. By learning algebra, students gain critical analytical skills. These are necessary for resolving equations and understanding the quantitative relationships presented in daily life.
In educational algebra:

• We learn to manipulate variables and understand how they interact in equations. This skill is vital in forming and solving both linear and quadratic equations.
• Algebraic equations describe a wide array of phenomena, making skills in algebra critical for various careers and fields.
• Concepts like solving quadratic equations often involve understanding complex calculations, but with a solid algebraic foundation, they become straightforward.

By engaging with algebra, students develop problem-solving skills that are applicable beyond mathematics, helping in logical reasoning and decision making.