Problem 89 A private school is going to sel... [FREE SOLUTION]

Chapter 3: Problem 89

A private school is going to sell raffle tickets as a fund raiser. Suppose the number $n$ of raffle tickets that will be sold is predicted by the equation $n=-20 p+300,$ where $p$ is the price of a raffle ticket in dollars. Graph the equation and use the graph to predict the number of raffle tickets that will be sold at a price of $\$ 6$.

Short Answer

Expert verified

At a price of $ \$6 $, 180 raffle tickets will be sold.

Step by step solution

Identify the Equation

The equation given is $ n = -20p + 300 $, where $ n $ is the number of raffle tickets sold and $ p $ is the price per ticket in dollars. This is a linear equation in slope-intercept form.

Understanding the Components of the Equation

In $ n = -20p + 300 $, the slope is $-20$, indicating that for every 1 dollar increase in the price $ p $, 20 fewer tickets are sold. The intercept is 300, meaning if the tickets were free ($ p = 0 $), 300 tickets would be sold.

Determine Key Points for Graph

To graph the equation, we can find key values by choosing different values of $ p $. For example, at $ p = 0 $, $ n = 300 $. For $ p = 15 $ (theoretically), $ n = 0 $ because $-20 \times 15 + 300 = 0 $. These points will help us plot the line.

Plot the Graph

Draw a coordinate plane with $ p $ on the x-axis and $ n $ on the y-axis. Plot the points $(0, 300)$ and $(15, 0)$ and draw a line through them. This line represents the equation $ n = -20p + 300 $.

Find the Number of Tickets Sold at $ p = 6 $

Using the graph, find $ p = 6 $ on the x-axis and locate the corresponding point on the line. This point shows the predicted number of tickets. Alternatively, substitute $ p = 6 $ directly into the equation: $ n = -20(6) + 300 = 180 $.

Interpret Results

The graph shows, and our calculation confirms, that at a price of $ \$6 $, the school will sell 180 raffle tickets. This can be seen where the line intersects the vertical line at $ p = 6 $.

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Key Concepts

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

Graphing Linear Equations

Graphing a linear equation is a method used to visualize relationships between variables. In the context of the equation $ n = -20p + 300 $, each point on the graph represents a combination of price $ p $ and the corresponding number of raffle tickets $ n $ that the school expects to sell. The purpose of graphing the equation is to provide a clear visual representation of how changes in price affect the number of tickets sold.

To graph a linear equation like this, follow these steps:

Identify key points: These are points where the equation is easy to solve, often at intercepts like $(0, 300)$ and $(15, 0)$. These calculations show that when the price of a ticket is $0, 300 tickets are sold, and when the price is $15, no tickets are sold.
Plot the points: Use a graph with $ p $ on the x-axis (horizontal) and $ n $ on the y-axis (vertical). Mark the points $(0, 300)$ and $(15, 0)$.
Draw the line: Use a ruler to connect the points. The resulting line is called the graph of the equation, showing the relation between $ p $ and $ n $.

Graphing helps us easily predict the outcome for any given ticket price within this range.

Slope-Intercept Form

The slope-intercept form is a specific way to express linear equations, written as $ y = mx + b $. Here, $ m $ is the slope and $ b $ is the y-intercept. In our exercise, the equation $ n = -20p + 300 $ follows this form:

The slope $ -20 $ tells us the rate of change. Specifically, it tells us that for every dollar increase in price $ p $, the number of tickets sold $ n $ decreases by 20. This negative relationship is common in scenarios where a higher price reduces demand.
The y-intercept $ 300 $ indicates the number of tickets sold when $ p $, the price, is zero. The graph of the equation would start on the y-axis at $ n = 300 $ if tickets were free, showing the maximum possible sales.

Understanding the slope and y-intercept gives insights into how sensitive sales are to price changes and the theoretical maximum sales possible.

Coordinate Plane

A coordinate plane is a two-dimensional space used to graph equations. It consists of two perpendicular lines called axes. The horizontal line is the x-axis (in this case, the price $ p $), and the vertical line is the y-axis (the number of tickets $ n $).

Each point on the plane corresponds to a pair $(x, y)$, providing a visual structure to analyze relationships between these variables. Using our equation case, here鈥檚 how it functions:

The x-axis represents possible prices of tickets, allowing us to see how different prices might impact sales.
The y-axis reflects the potential number of tickets sold at those prices, aligning with the predictions of the linear equation.
By plotting points from the equation onto the coordinate plane, we create a line that visually connects all solutions $(p, n)$ of the equation.

Through the coordinate plane, complex relationships become easier to interpret, revealing trends and patterns at a glance.

91影视

Short Answer

Step by step solution

Identify the Equation

Understanding the Components of the Equation

Determine Key Points for Graph

Plot the Graph

Find the Number of Tickets Sold at \( p = 6 \)

Interpret Results

Key Concepts

Graphing Linear Equations

Slope-Intercept Form

Coordinate Plane

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