Problem 20 - Calculus Concepts

Question

Determine whether there is a specific market price above which demand is zero or price per unit is unbounded. Write the maximum possible market price, using dollars per unit as the units of measure for input. \(D(p)=-p^{2}+49\) units

Step-by-Step Solution

Answer

The maximum market price is $7 per unit before demand becomes zero.

Step 1: Interpret the Demand Function

We are given the demand function \( D(p) = -p^2 + 49 \). This function describes how the demand for a product (\(D\)) changes with respect to its price (\(p\)). A negative coefficient for \(p^2\) indicates that the function is a downward-facing parabola.

Step 2: Find the Price at Which Demand is Zero

The demand will be zero when \( D(p) = 0 \). Set the equation \( -p^2 + 49 = 0 \) and solve for \( p \).

Step 3: Solve the Quadratic Equation

Rearrange the equation \( -p^2 + 49 = 0 \) to \( p^2 = 49 \). Solving for \( p \), we take the square root of both sides: \( p = \pm\sqrt{49} \), which gives \( p = 7 \) or \( p = -7 \). Since price cannot be negative, \( p = 7 \) is the only practical solution.

Step 4: Interpret the Solution

The maximum theoretical market price where demand drops to zero is \(7\) dollars per unit. Beyond this price, the demand will no longer exist for this product.

Key Concepts

Quadratic Equation

Quadratic equations are an important concept in algebra and are used to model a variety of real-world phenomena, including economic behaviors such as demand functions. A quadratic equation is typically expressed in the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. The presence of \( x^2 \) means that the relationship described is not linear, but rather curved. This curvature is crucial when analyzing how demand changes with price.

The given demand function \( D(p) = -p^2 + 49 \) is a classic example of a quadratic equation, where it predicts how the demand \( D \) for a product will decrease as the price \( p \) increases. In this function, the coefficient \(-1\) of \( p^2 \) ensures a downward-opening parabola, indicating that as price moves away from its optimal point, demand will still decrease.

Market Price

Market price refers to the current price at which an asset or service can be bought or sold. In economics, understanding how prices relate to demand is crucial, as it helps businesses set prices that maximize revenue and manage inventory. Market prices result from the interaction of supply and demand in a marketplace.

In the context of the given demand function, market price is represented by \( p \). The function \( D(p) = -p^2 + 49 \) illustrates the relationship between price and the quantity demanded by consumers. When prices increase, consumer interest decreases, leading to a reduction in demand. Here, the task is to find the specific market price above which demand becomes zero. In simpler terms, determining the maximum price that consumers are willing to pay before opting out of a purchase entirely.

Parabola

A parabola is a symmetrical, U-shaped curve that can open either upwards or downwards, depending on the sign of the coefficient of its squared term. When dealing with quadratic equations, the graph of the equation represents a parabola. This is important in understanding demand functions because it visually represents how demand will change over a range of prices.

In the provided demand function \( D(p) = -p^2 + 49 \), the negative sign before \( p^2 \) creates a downward-opening parabola. This indicates that as price \( p \) moves from zero to higher values, the demand decreases, forming an inverted 'U' shape on the graph. The peak of this parabola can be particularly insightful, as it indicates the maximum demand prior to declines as prices continue to rise.

Price-Demand Relationship

The price-demand relationship is a foundational principle in economics that describes how changes in price affect the quantity demanded. A fundamental understanding is that typically, as price increases, the quantity demanded decreases, and vice versa, illustrating the law of demand.

Analyzing the demand function \( D(p) = -p^2 + 49 \), we observe this relationship clearly. When \( p = 0 \), demand is at its highest because there is no cost to the consumer. As \( p \) increases, demand decreases. When the price reaches \( p = 7 \), demand becomes zero, indicating that consumers are no longer willing to buy the product. The equation and its resulting parabola graphically demonstrate this decreasing relationship, helping us understand consumer behavior in response to changes in market prices.