A car rental agency is considering a modification in its oil change procedure. Currently, it uses a Type X filter, which costs $5 and must be changed every 9 comma 000 miles along with the oil (5 quarts). Between each oil change, one quart of oil must be added after each 500 miles. The proposed filter (Type Y) has to be replaced every 7 comma 000 miles (along with 5 quarts of oil) but does not require any additional oil between filter changes. If the oil costs $1.09 per quart, what is the maximum acceptable price for the Type Y filter?

We know that the X filter costs $5 and must be changed every 9,000 miles, with the oil 5 quartz, and one quart of oil must be added every 500 miles, the resultant equation for the price of those 9,000 miles is:

[tex]\frac{5+ 5(1.09)+16(1.09)}{9000}[/tex]

And to express the new oil it would be:

[tex]\frac{x+ 5(1.09)+}{7000}[/tex]

So we just have to equalize them:

[tex]\frac{x+ 5(1.09)+}{7000}=\frac{5+ 5(1.09)+16(1.09)}{9000}\\9000x+49,050=195,230\\9000x=195,230-49,050=146,180\\x=\frac{146180}{9000} \\x= 16.2422[/tex]

So we know that the maximum price that the new oil filter would be $16.2422 because at that point it costs the same as the other filter witht he cost of the adding of the oil quarts.