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Answer :
We are given two points on the profit line:
- When [tex]$x = 60$[/tex] magazines, the profit is [tex]$y = 220$[/tex] dollars.
- When [tex]$x = 100$[/tex] magazines, the profit is [tex]$y = 420$[/tex] dollars.
Step 1. Calculate the Slope
The slope [tex]$m$[/tex] of a line through two points [tex]$\left(x_1,y_1\right)$[/tex] and [tex]$\left(x_2,y_2\right)$[/tex] is given by
[tex]$$
m = \frac{y_2 - y_1}{x_2 - x_1}.
$$[/tex]
Substitute the given values:
[tex]$$
m = \frac{420 - 220}{100 - 60} = \frac{200}{40} = 5.
$$[/tex]
Step 2. Write the Equation in Point-Slope Form
Using the point-slope form of a line,
[tex]$$
y - y_1 = m(x - x_1),
$$[/tex]
we take the point [tex]$\left(60, 220\right)$[/tex] and substitute [tex]$m=5$[/tex]:
[tex]$$
y - 220 = 5(x - 60).
$$[/tex]
This is the equation that models the total profit [tex]$y$[/tex] based on the number of magazines sold [tex]$x$[/tex].
Step 3. Identify the Correct Option
Comparing with the provided options, we see that the equation
[tex]$$
y - 220 = 5(x - 60)
$$[/tex]
corresponds to option D.
Thus, the correct answer is option D.
- When [tex]$x = 60$[/tex] magazines, the profit is [tex]$y = 220$[/tex] dollars.
- When [tex]$x = 100$[/tex] magazines, the profit is [tex]$y = 420$[/tex] dollars.
Step 1. Calculate the Slope
The slope [tex]$m$[/tex] of a line through two points [tex]$\left(x_1,y_1\right)$[/tex] and [tex]$\left(x_2,y_2\right)$[/tex] is given by
[tex]$$
m = \frac{y_2 - y_1}{x_2 - x_1}.
$$[/tex]
Substitute the given values:
[tex]$$
m = \frac{420 - 220}{100 - 60} = \frac{200}{40} = 5.
$$[/tex]
Step 2. Write the Equation in Point-Slope Form
Using the point-slope form of a line,
[tex]$$
y - y_1 = m(x - x_1),
$$[/tex]
we take the point [tex]$\left(60, 220\right)$[/tex] and substitute [tex]$m=5$[/tex]:
[tex]$$
y - 220 = 5(x - 60).
$$[/tex]
This is the equation that models the total profit [tex]$y$[/tex] based on the number of magazines sold [tex]$x$[/tex].
Step 3. Identify the Correct Option
Comparing with the provided options, we see that the equation
[tex]$$
y - 220 = 5(x - 60)
$$[/tex]
corresponds to option D.
Thus, the correct answer is option D.
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