We have the relationship between two variables that can be modeled by a line equation. We need first to determine the equation of the line. To do that can use the following equation for a line:
[tex]y=mx+b[/tex]Where "m" is the slope and b the y-intercept. To determine the slope "m" we use the following formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Taking two points in the line:
[tex]\begin{gathered} (x_1,y_1)=(2,246) \\ (x_2,y_2)=(4,450) \end{gathered}[/tex]Replacing in the formula for the slope:
[tex]m=\frac{450-246}{4-2}[/tex]Solving the operations:
[tex]m=\frac{204}{2}=102[/tex]replacing in the equation for the line:
[tex]y=102x+b[/tex]To find the value of "b" we replace any point through which the line passes. Replacing (x,y) = (2,246):
[tex]246=102(2)+b[/tex]Solving the product:
[tex]246=204+b[/tex]subtracting 204 to both sides:
[tex]\begin{gathered} 246-204=b \\ 42=b \end{gathered}[/tex]Replacing in the line equation:
[tex]y=102x+42[/tex]The processing fee would be the initial fee and in the line equation, it is equivalent to the y-intercept therefore, the processing fee is $42. The daily fee is equivalent to the slope of the line, therefore, the daily fee is $102 per day.
If the total amount to spend is $1200 then replacing in the line equation:
[tex]1200=102x+42[/tex]Solving for "x" first by subtracting 42 to both sides:
[tex]\begin{gathered} 1200-42=102x \\ 1158=102x \end{gathered}[/tex]Dividing both sides by 102:
[tex]\begin{gathered} \frac{1158}{102}=x \\ 11.4=x \end{gathered}[/tex]Therefore, the number of days is 11.4.
