a We have been given a table showing the cost for renting a beach house.
Days
2
4
6
8
Total Cost (dollars)
246
450
546
858
The situation can be modeled by a linear equation if the rate of change is constant. We can calculate the rate of change using the Slope Formula. m=y_2-y_1/x_2-x_1
We can see that the rate of change is constant. Therefore, the situation can be modeled by a linear equation.
b To determine the processing fee and the daily fee, let's find the linear equation which models this situation. Recall the point-slope form of a linear equation.
y- y_1= m(x- x_1)
Here m is the rate of change and ( x_1, y_1) is a point on the line. In Part A we already determined that the rate of change is 102. Let's substitute this and the point ( 2, 246) into the formula.
y- y_1&= m(x- x_1)
&⇓
y- 246&= 102(x- 2)
To find the processing fee and the daily fee we need to rewrite this into slope-intercept form.
We now have an equation which models the situation.
y=102x+42
Now we know the daily fee is $102. The processing fee is $42, as this is a fixed amount that is paid only once no matter how many days the beach house is rented.
c Since we can spend no more than $1200 on the beach house rental, the cost must be less than or equal to 1200. We can represent this as an inequality.
Since x is the number of days and we cannot rent the house for a fraction of a day, we can conclude that the maximum number of days we can rent the beach house is 11.
