Answer :
We know the relationship between the cost and number of cups is linear. We also know that the equation of a linear relationship is:
[tex]y-y_1=m(x-x_1)[/tex]where m is the slope and (x1,y1) is a point on the line.
In this problem we know that the cost is represented by the variable y and the number of cups is represented by the variable x.
This means that we can relate the number of cups and cost as points in the plane. With this in mind we have the points (100,61) and (300,161). Once we have two points we can obtained the slope using the expression:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]in this case:
[tex]\begin{gathered} m=\frac{161-61}{300-100} \\ =\frac{100}{200} \\ =\frac{1}{2} \end{gathered}[/tex]Once we have the slope we can obtained the equation:
[tex]\begin{gathered} y-61=\frac{1}{2}(x-100) \\ y-61=\frac{1}{2}x-50 \\ y=\frac{1}{2}x+11 \end{gathered}[/tex]Therefore, the equation that represents the cost in function of the number of cups is:
[tex]y=0.5x+11[/tex]Once we have the equation we can know how many cups are produced if the cost is 236. To do this we plug y=236 in the equation and solve for x:
[tex]\begin{gathered} 236=0.5x+11 \\ 0.5x=225 \\ x=\frac{225}{0.5} \\ x=450 \end{gathered}[/tex]Therefore, if the cost of production is $236 the restaurant made 450 cups of coffee.