Answer :
Sure! Let's work through the problem step by step to find which equation models the total profit, [tex]\( y \)[/tex], based on the number of magazines sold, [tex]\( x \)[/tex].
We have two data points for the number of magazines sold and their corresponding profit:
1. When 60 magazines are sold, the profit is \[tex]$220.
2. When 100 magazines are sold, the profit is \$[/tex]420.
From these two points, we can determine the linear relationship between the number of magazines sold and the profit.
Step 1: Calculate the slope ([tex]\( m \)[/tex])
The slope of a line ([tex]\( m \)[/tex]) is calculated using the formula:
[tex]\[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \][/tex]
In our case:
- [tex]\( x_1 = 60 \)[/tex], [tex]\( y_1 = 220 \)[/tex]
- [tex]\( x_2 = 100 \)[/tex], [tex]\( y_2 = 420 \)[/tex]
Substituting these values into the formula gives:
[tex]\[ m = \frac{{420 - 220}}{{100 - 60}} = \frac{200}{40} = 5 \][/tex]
Step 2: Write the equation in point-slope form
The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using [tex]\( m = 5 \)[/tex] and the point [tex]\( (x_1, y_1) = (60, 220) \)[/tex], the equation becomes:
[tex]\[ y - 220 = 5(x - 60) \][/tex]
This equation represents the total profit [tex]\( y \)[/tex] based on the number of magazines sold [tex]\( x \)[/tex].
Conclusion:
The correct equation that models the total profit based on the number of magazines sold is:
[tex]\[ y - 220 = 5(x - 60) \][/tex]
Thus, the correct option is D.
We have two data points for the number of magazines sold and their corresponding profit:
1. When 60 magazines are sold, the profit is \[tex]$220.
2. When 100 magazines are sold, the profit is \$[/tex]420.
From these two points, we can determine the linear relationship between the number of magazines sold and the profit.
Step 1: Calculate the slope ([tex]\( m \)[/tex])
The slope of a line ([tex]\( m \)[/tex]) is calculated using the formula:
[tex]\[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \][/tex]
In our case:
- [tex]\( x_1 = 60 \)[/tex], [tex]\( y_1 = 220 \)[/tex]
- [tex]\( x_2 = 100 \)[/tex], [tex]\( y_2 = 420 \)[/tex]
Substituting these values into the formula gives:
[tex]\[ m = \frac{{420 - 220}}{{100 - 60}} = \frac{200}{40} = 5 \][/tex]
Step 2: Write the equation in point-slope form
The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using [tex]\( m = 5 \)[/tex] and the point [tex]\( (x_1, y_1) = (60, 220) \)[/tex], the equation becomes:
[tex]\[ y - 220 = 5(x - 60) \][/tex]
This equation represents the total profit [tex]\( y \)[/tex] based on the number of magazines sold [tex]\( x \)[/tex].
Conclusion:
The correct equation that models the total profit based on the number of magazines sold is:
[tex]\[ y - 220 = 5(x - 60) \][/tex]
Thus, the correct option is D.
