Problem 99 - Algebra A Combined Function

Question

\(A\) common equation used in business is a demand equation. It expresses the relationship between the unit price of some commodity and the quantity demanded. For Exercises 99 and 100, \(p\) represents the unit price and \(x\) represents the quantity demanded in thousands. A manufacturing company has found that the demand equation for a certain type of scissors is given by the equation \(p=-x^{2}+47\). Find the demand for the scissors if the price is \(\$ 11\) per pair.

Step-by-Step Solution

Answer

The demand for scissors is 6000 pairs.

Step 1: Understand the equation

The given demand equation is \( p = -x^2 + 47 \). Here, \( p \) represents the price per pair of scissors and \( x \) represents the quantity demanded in thousands.

Step 2: Substitute price into the equation

We are given \( p = 11 \). Substitute 11 into the equation for \( p \) to find \( x \). The equation becomes: \( 11 = -x^2 + 47 \).

Step 3: Solve for \( x^2 \)

Rearrange the equation to solve for \( x^2 \): \( 11 = -x^2 + 47 \) becomes \( -x^2 = 11 - 47 \). This simplifies to \( -x^2 = -36 \).

Step 4: Isolate \( x^2 \)

Divide both sides of the equation by \(-1\) to isolate \( x^2 \): \( x^2 = 36 \).

Step 5: Solve for \( x \)

Take the square root of both sides to find \( x \): \( x = \sqrt{36} \). This gives \( x = 6 \).

Step 6: Interpret the result

Since \( x \) represents the quantity demanded in thousands, a solution of \( x = 6 \) means that the demand for scissors is 6000 pairs.

Key Concepts

Quadratic Equations

A quadratic equation is an equation that can be written in the form \( ax^2 + bx + c = 0 \). The quadratic equation we are discussing here is derived from the demand function where the equation is \( p = -x^2 + 47 \). In this equation, \( p \) is the price, and \( x \) is the quantity demanded. The most distinguishing feature of a quadratic equation is the term \( x^2 \), which gives the equation its parabolic shape when graphed. Quadratic equations can have two real solutions, one real solution, or no real solution at all, depending on the discriminant (\( b^2 - 4ac \)). In business contexts, these solutions represent real-world quantities or values like prices or quantities demanded, as seen in this exercise.Key characteristics of quadratic functions and their graphs include:

• The direction of the parabola (upward if \( a > 0 \), downward if \( a < 0 \)).
• The vertex, which is the highest or lowest point on the graph, crucial for understanding maximum and minimum values.
• The roots, where the parabola intersects the x-axis, representing the values that satisfy the equation when \( p = 0 \).

Understanding these characteristics is fundamental in solving problems involving quadratic equations.

Solving Equations

Solving equations is a critical skill in mathematics, as it involves finding the unknown values that satisfy the given equation. In this particular problem, we solved a demand equation where \( p = 11 \) was substituted into the quadratic equation \( 11 = -x^2 + 47 \). This step is crucial because substituting known values allows us to solve for the unknown.The process involves several clear steps:

• Substitute the known values into the equation.
• Rearrange or simplify the equation to isolate the variable.
• Perform algebraic operations, such as addition and division, to solve for the unknown.
• Take necessary roots or powers, as in this case where we take the square root to solve for \( x \).

Finding the correct solution, \( x = 6 \), involves ensuring all arithmetic steps align correctly to yield the true value of \( x \). Each of these steps requires logical reasoning and attention to mathematical details to arrive at the correct conclusion.

Business Mathematics

Business mathematics involves applying mathematical principles to solve business problems. It plays a vital role in various business operations, such as pricing, profits, and demand-supply analysis. In this exercise, we see an example of a demand equation, a common concept in business mathematics. Demand equations help businesses understand how different prices might affect the quantity demanded of their products. Important concepts in business mathematics tied to demand equations include:

• Price elasticity: Determines how sensitive the quantity demanded is to a change in price.
• Revenue calculations: As derived from multiplying price by quantity demanded, assisting in forecasting sales and profits.
• Break-even analysis: Identifying the point at which total cost and total revenue are equal, helping businesses make critical pricing and production decisions.

Understanding and using these calculations effectively empower businesses to make informed decisions, influencing strategies such as pricing and production to maximize profit and efficiency. In essence, business mathematics provides the quantitative foundation for strategic decision-making in business environments.