Given that for each $2 increase in price, the demand is less and 4 fewer cars are rented.
Let x be the number of $2 increases in price, then the revenue from renting cars is given by
[tex](48 + 2x) \times (150 - 4x)=7,200+108x-8x^2[/tex].
Also, given that for each car that is rented, there are routine maintenance costs of $5 per day, then the total cost of renting cars is given by
[tex]5(150-4x)=750-20x[/tex]
Profit is given by revenue - cost.
Thus, the profit from renting cars is given by
[tex](7,200+108x-8x^2)-(750-20x)=6,450+128x-8x^2[/tex]
For maximum profit, the differentiation of the profit function equals zero.
i.e.
[tex] \frac{d}{dx} (6,450+128x-8x^2)=0 \\ \\ 128-16x=0 \\ \\ x= \frac{128}{16} =8[/tex]
The price of renting a car is given by 48 + 2x = 48 + 2(8) = 48 + 16 = 64.
Therefore, the rental charge will maximize profit is $64.
