A car rental agency rents 190 cars per day at a rate of 29 dollars per day. for each 1 dollar increase in the daily rate, 5 fewer cars are rented. at what rate should the cars be rented to produce the maximum income, and what is the maximum income?

The equation that we can create from this situation is:

i = (190 - 5 x) * (29 + x)

where i is the income and x is the increase in daily rate

Expanding the equation:

i = 5510 + 190x - 145x - 5x^2

i = - 5x^2 + 45x + 5510

Taking the 1st derivative:

di/dx = - 10x + 45

Set to zero to get the maxima:

-10x + 45 = 0

x = 4.5

So the cars should be rented at:

29 + x = 33.5 dollars per day

The maximum income is:

i = (190 - 5*4.5) * (33.5)

i = 5,611.25 dollars