Problem 68 The manager of a weekend flea ma... [FREE SOLUTION]

Chapter 1: Problem 68

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?

Short Answer

Expert verified

Slope: decrease of 4 spaces per $ increase; y-intercept: 200 spaces at $0; x-intercept: 0 spaces at $50.

Step by step solution

Identify Equation Form

The given equation is a linear equation of the form $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept. For the equation $ y = 200 - 4x $, the slope $ m = -4 $ and the $ y $-intercept $ b = 200 $.

Determine and Plot Intercepts

To find the $ y $-intercept, set $ x = 0 $: $ y = 200 - 4(0) = 200 $. So, the point is (0, 200). For the $ x $-intercept, set $ y = 0 $: $ 0 = 200 - 4x $, solving gives $ x = 50 $. So, the point is (50, 0). Plot the points (0, 200) and (50, 0) on a graph.

Draw the Line

Using the intercepts from Step 2, draw the line that connects them on the graph. Make sure the line only covers the region where both $ x $ and $ y $ are non-negative.

Interpret the Slope

The slope $ m = -4 $ indicates that for each additional dollar charged per space, the number of spaces rented decreases by 4.

Interpret the Y-Intercept

The $ y $-intercept $ b = 200 $ indicates that if the rental charge is $ 0 $ dollars (free), the market rents out 200 spaces.

Interpret the X-Intercept

The $ x $-intercept occurs at $ x = 50 $, which means that charging \$50 per space results in renting 0 spaces.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Interpretation

In a linear function like the one provided, the slope signifies the rate of change between two variables. Here, the slope is represented by the coefficient of the variable $ x $, which is $-4$. This means that for every additional dollar increase in the rental charge per space, the number of spaces rented decreases by four.
Think of the slope as a measure of responsiveness: how much does one variable change when the other one changes?

A negative slope indicates an inverse relationship鈥攈ere, as the price increases, fewer spaces are rented.
A slope of $-4$ tells us the exact amount by which the rental quantity declines per dollar increase.

Visualizing this on a graph, the line will tilt downward from left to right, reinforcing this inverse relationship.

Y-Intercept

The $ y $-intercept of a linear function corresponds to the point where the line crosses the $ y $-axis. In this context, it is represented by the value $ b $ in the equation $ y = mx + b $.
For the given equation $ y = 200 - 4x $, the $ y $-intercept is $ 200 $, which we determine by setting $ x $ to zero.

This intercept, (0, 200), indicates that if no charge is made for the rental spaces, all 200 spaces are rented out.
It provides an intuitive baseline where the effect of pricing is completely removed.

Thus, the $ y $-intercept reflects the maximum capacity of spaces rented out at no cost.

X-Intercept

The $ x $-intercept of a line is the point where the graph crosses the $ x $-axis. It signifies where the output $ y $ equals zero. To find this point, you set $ y $ to zero and solve for $ x $.
In our equation $ y = 200 - 4x $, setting $ y $ to zero gives us $ x = 50 $.

This point, (50, 0), indicates the rental price at which no spaces are rented out鈥攅ssentially the price limit for this situation.
The $ x $-intercept provides a valuable threshold in budgeting or pricing strategies.

It shows where the cost becomes prohibitive and all demand is lost.

Graphing Linear Equations

Graphing linear equations involves plotting points and drawing a line through them. For an equation in the form $ y = mx + b $, start by identifying the $ y $-intercept and $ x $-intercept. These intercepts are the easiest to plot because they lie on the axes.
In our scenario:

Plot the $ y $-intercept at $ (0, 200) $.
Plot the $ x $-intercept at $ (50, 0) $.

Connect these two points with a straight line, ensuring the line extends within the limits where both sales price and spaces are non-negative.
This linear graph visually captures the relationship, showing how changes in price affect the number of rented spaces.

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Short Answer

Step by step solution

Identify Equation Form

Determine and Plot Intercepts

Draw the Line

Interpret the Slope

Interpret the Y-Intercept

Interpret the X-Intercept

Key Concepts

Slope Interpretation

Y-Intercept

X-Intercept

Graphing Linear Equations

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