Members of the student council are conducting a fundraiser by selling school calendars. After selling 80 calendars, they had a loss of $360. After selling 200 calendars, they had a profit of $600. Write an equation that describes the relation between the profit or loss, and x, the number of calendars sold. How much profit did they make from selling each calendar? How much would they have lost if they had sold no calendars?
The correct answer and explanation is :
To solve this problem, we need to establish an equation that links the profit or loss to the number of calendars sold, denoted by ( x ). Let’s break it down step by step.
Step 1: Setting up the Equation
The general form of a linear equation is:
[
P(x) = mx + b
]
where:
- ( P(x) ) is the profit (or loss) from selling ( x ) calendars,
- ( m ) is the slope of the line, representing the profit (or loss) per calendar,
- ( b ) is the y-intercept, which represents the profit or loss when no calendars are sold (i.e., the fixed cost).
We are given two key pieces of information:
- When 80 calendars are sold, there is a loss of $360.
- When 200 calendars are sold, there is a profit of $600.
This gives us two points on the line:
- ( (80, -360) ),
- ( (200, 600) ).
Step 2: Finding the Slope
To find the slope (( m )), we use the formula for the slope between two points:
[
m = \frac{y_2 – y_1}{x_2 – x_1}
]
where ( (x_1, y_1) = (80, -360) ) and ( (x_2, y_2) = (200, 600) ).
Substitute the values into the formula:
[
m = \frac{600 – (-360)}{200 – 80} = \frac{600 + 360}{120} = \frac{960}{120} = 8
]
Thus, the slope of the line, ( m ), is 8. This means they make a profit of $8 per calendar sold.
Step 3: Finding the Y-Intercept
Now, we use one of the points to find the y-intercept (( b )). Using the point ( (80, -360) ) and the slope ( m = 8 ), we substitute into the equation:
[
P(80) = 8(80) + b
]
[
-360 = 640 + b
]
[
b = -360 – 640 = -1000
]
Step 4: Writing the Equation
Now that we have the slope ( m = 8 ) and the y-intercept ( b = -1000 ), the equation that describes the relation between the profit and the number of calendars sold is:
[
P(x) = 8x – 1000
]
Step 5: Interpretation
- The profit made from selling each calendar is $8 (the slope of the line).
- If no calendars are sold (( x = 0 )), the loss would be ( P(0) = 8(0) – 1000 = -1000 ). Therefore, they would have lost $1000 if no calendars were sold, representing the fixed costs or initial expenses.
Conclusion
The equation that models the profit or loss from selling ( x ) calendars is ( P(x) = 8x – 1000 ). Selling each calendar results in a profit of $8, and if no calendars are sold, the loss would be $1000.