To determine the equation that models the total profit, \( y \), based on the number of hotdogs sold, \( x \), we need to find the equation of a line that passes through the given points \((40, 90)\) and \((80, 210)\).
1. Find the Slope (\( m \)) of the Line:
The slope \( m \) is calculated using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Given points \((x_1, y_1) = (40, 90)\) and \((x_2, y_2) = (80, 210)\), plug these values into the slope formula:
[tex]\[
m = \frac{210 - 90}{80 - 40} = \frac{120}{40} = 3
\][/tex]
So, the slope \( m \) is 3.
2. Use the Point-Slope Form of the Equation of a Line:
The point-slope form of a line's equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Using the point \((40, 90)\) and the slope \( m = 3 \), we substitute these values into the point-slope form:
[tex]\[
y - 90 = 3(x - 40)
\][/tex]
3. Select the Correct Equation:
Comparing the options given:
[tex]\[
\begin{align*}
A. & \quad y + 90 = 3(x + 40) \\
B. & \quad y - 90 = 2.6(x - 40) \\
C. & \quad y + 90 = 2.6(x + 40) \\
D. & \quad y - 90 = 3(x - 40)
\end{align*}
\][/tex]
We see that the correct equation that matches our derived equation \( y - 90 = 3(x - 40) \) is D.
Therefore, the equation that models the total profit, \( y \), based on the number of hotdogs sold, \( x \), is:
[tex]\[ \boxed{y - 90 = 3(x - 40)} \][/tex]
