Problem 52 - Elementary Algebra

Question

The marketing department at a large company has been able to express the relationship between the demand for a product and its price by using statistical techniques. The department found, by analyzing studies done in six different market areas, that the equation giving the approximate demand for a product (in thousands of units) for a particular price (in cents) is \(y=-14.15 x+257.11\). Find the approximate number of units demanded when the price is a. \(\$ 0.12\) b. \(\$ 0.15\)

Step-by-Step Solution

Answer

Question: Determine the approximate number of units demanded when the price is (a) $0.12 and (b) $0.15, given the demand equation \(y = -14.15x + 257.11\), where \(y\) represents the demand (in thousands of units) and \(x\) represents the price (in cents). Answer: a. The approximate number of units demanded when the price is $0.12 is approximately 256.29 thousand units. b. The approximate number of units demanded when the price is $0.15 is approximately 250.36 thousand units.

Step 1: In the given question, the prices provided are in dollars. But we require the price to be in cents for the formula to work. So, let's convert the given prices into cents first: a. \(\$ 0.12 = 12 \text{ cents}.\) b. \(\$ 0.15 = 15 \text{ cents}.\) #Step 2: Substitute the given prices in the demand equation#

Now we need to substitute the given prices in cents into the demand equation, \(y=-14.15 x+257.11\), to find the approximate number of units demanded. #Step 3: Find the approximate number of units demanded for \(\ 12\) cents#

Step 2: Let's substitute \(x = 12\) in the demand equation: \(y = -14.15 (12) + 257.11\) \(y \approx 256.29\) Therefore, the approximate number of units demanded when the price is \(\$ 0.12\) is approximately \(256.29\) thousand units. #Step 4: Find the approximate number of units demanded for \(\ 15\) cents#

Now, we need to substitute \(x = 15\) in the demand equation: \(y = -14.15 (15) + 257.11\) \(y \approx 250.36\) So, the approximate number of units demanded when the price is \(\$ 0.15\) is approximately \(250.36\) thousand units.

Key Concepts

Linear Equations

Understanding linear equations is fundamental to grasp the algebra behind demand equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are simple to recognize because of their distinctive format. The general formula for a linear equation in two variables is given by \( y = mx + b \), where \( m \) is the slope, \( b \) is the y-intercept, and \( x \) and \( y \) are variables. The slope, \( m \), represents the rate of change, showing how much \( y \) changes with a one-unit increase in \( x \) and the y-intercept, \( b \) indicates the value of \( y \) when \( x \) is zero.

When working with the demand equation \( y = -14.15x + 257.11 \), \( y \) represents the demand in thousands of units, \( x \) stands for the price in cents, the slope of \( -14.15 \) reflects how demand decreases with an increase in price, and \( 257.11 \) is the demand when the price is zero cents. To solve a linear equation with a given \( x \) value, simply multiply the slope by the \( x \) value, and add the y-intercept, as demonstrated in the textbook solution.

Graphing Linear Equations

Graphing linear equations visually represents the relationship between two quantities. It can also provide insight into the price-demand relationship of a product, showcasing how changes in price affect demand. To graph a linear equation like \( y = -14.15x + 257.11 \), you begin by plotting the y-intercept on the y-axis. Here the y-intercept is \( 257.11 \), representing the point where the line will cross the y-axis, which is where \( x = 0 \).

Next, you use the slope, which in this case is \( -14.15 \), to determine how to draw the line. Since the slope is negative, the line will fall as it moves from left to right, indicating a negative correlation between price and demand. You can plot a second point by moving \( 14.15 \) units down for every 1 unit you move to the right on the graph. Drawing a straight line through both points will give the graph of the demand equation. The steepness of this line conveys how sensitive the demand for the product is to price changes, a concept known as price elasticity.

Price-Demand Relationship

The price-demand relationship in economics is an important concept that can be captured through algebra and graphing. It refers to the connection between the price of a product and the quantity of that product customers are willing to buy. Typically, the demand decreases as the price increases, as reflected by a downward sloping line in a graph. This inverse relationship is also neatly summarized in algebraic terms by the demand equation, brought to life in our example of \( y = -14.15x + 257.11 \).

In this equation, a negative sign precedes the coefficient of \( x \), indicating that the price and demand are inversely related: higher prices lead to lower demand. This inverse relationship is a cornerstone in understanding market dynamics. In the provided exercise, for instance, as the price increases from \( 0.12 \) dollars (or \( 12 \) cents) to \( 0.15 \) dollars (or \( 15 \) cents), demand for units drops from \( 256.29 \)-thousand units to \( 250.36 \)-thousand units, illustrating how the demand decreases with higher prices. It's essential for students to not only calculate these numbers but also internalize the economic theory behind them for a deeper understanding.