Problem 68 - Algebra and Trigonometry

Question

Flea Market The manager of a weekend flea market knows from past experience that if she charges \(x\) dollars for a rental space at the flea market, then the number \(y\) of spaces she can rent is given by the equation \(y=200-4 x\) (a) Sketch a graph of this linear equation. (Remember that the rental charge per space and the number of spaces rented must both be nonnegative quantities.) (b) What do the slope, the \(y\) -intercept, and the \(x\) -intercept of the graph represent?

Step-by-Step Solution

Answer

The slope is -4, indicating 4 fewer rentals per dollar increase. Y-intercept is 200 (maximum rentals at $0), and the x-intercept is 50 (price when rentals drop to zero).

Step 1: Understand the equation

The equation provided is \(y = 200 - 4x\), where \(y\) is the number of spaces rented and \(x\) is the rental cost per space. It's a linear equation in the form \(y = mx + b\).

Step 2: Identify the Slope

The coefficient of \(x\), which is \(-4\), is the slope of the line. This means for every additional dollar charged, 4 fewer spaces are rented.

Step 3: Determine the y-intercept

The y-intercept is the constant term, which is 200. This represents the maximum number of spaces rented if the cost per space is \($0\).

Step 4: Determine the x-intercept

To find the x-intercept, set \(y = 0\), giving the equation \(0 = 200 - 4x\). Solving for \(x\) gives \(x = 50\). This means if the cost per space is \($50\), no spaces will be rented.

Step 5: Graph the equation

Plot the y-intercept (0, 200) and the x-intercept (50, 0) on a graph. Draw a line through these two points to represent the equation. Ensure the line only includes when both x and y are nonnegative.

Step 6: Interpret the Graph

The graph shows the relationship between the rental price per space and the number of spaces rented. The slope shows the rate of decrease in rented spaces as the price increases. The y-intercept shows the maximum spaces rented when the price is zero, and the x-intercept shows the maximum price before rental becomes zero.

Key Concepts

Graph Interpretation

Graphing a linear equation gives us a visual representation of the relationship between the variables involved. In the equation \(y = 200 - 4x\), \(y\) is the number of flea market spaces rented, and \(x\) is the rental cost per space. Plotting this on a graph involves marking two key points: the y-intercept and x-intercept. From these points, you can draw a straight line representing all possible combinations of \(x\) and \(y\) that satisfy the equation.

To start, label your axes where the horizontal axis (x-axis) represents the rental cost, and the vertical axis (y-axis) represents the number of spaces. Plot the y-intercept (0, 200) and the x-intercept (50, 0). Connect these points with a straight line. The slope of the line will be negative due to the negative relationship between price and rental quantity.

• Each point on the line represents a potential cost per space and its corresponding number of rented spaces.
• The line will slope downwards to the right, showing that as price increases, the number of rentals decreases.

Slope of a Line

Understanding the slope of a line is crucial when dissecting linear equations. In this scenario, the slope is the coefficient of \(x\) in the equation \(y = 200 - 4x\), which is \(-4\). The slope of \(-4\) indicates how the number of rented spaces changes with the rental cost.

What the Slope Represents
The slope value of \(-4\) means that for every increase in the price by one dollar, the number of rented spaces decreases by four. This is a constant rate of change. Graphically, this slope steers the direction of the line drawn through the x and y-intercepts, guiding it downward as cost increases.

• A negative slope signifies a decrease in one variable as the other increases.
• This informs pricing strategies by highlighting the impact of price on demand.

X-intercept

The x-intercept of a graph provides insight into at which point one variable will reach zero when graphed against another. For the equation \(y = 200 - 4x\), the x-intercept tells us the maximum price at which no more flea market spaces will be rented.

Finding the X-intercept
To find this point, set \(y = 0\) and solve for \(x\): \[0 = 200 - 4x \] \[x = 50 \]
This means that if \(x\), the rental cost, is $50, all 200 spaces will remain unrented, capturing the highest possible price threshold before demand drops to zero.

• The x-intercept provides a financial metric to avoid overstretching rental prices.
• It gives clarity on price limitations and market thresholds.

Y-intercept

The y-intercept of a graph is a valuable point showing where the line crosses the y-axis. It indicates the initial value of the dependent variable (y) when the independent variable (x) is zero. In the flea market equation, this point is vital in understanding the base level of activity without any cost interference.

Understanding the Y-intercept
Here, when \(x = 0\), \(y = 200\). This means if the rental cost per space is \($0\), all 200 spaces will be rented. It reflects the maximum renting capacity at zero charge.

• The y-intercept shows potential when affordability is not a barrier.
• Useful for gauging market saturation and capacity utilization.

The y-intercept guides strategic decision-making around pricing and promotions, by determining the base market demand.