Let's first determine the relationship between the rental rate and the number of cars rented.
For each $1 increase in rate, 5 fewer cars are rented. Therefore, we can write the equation:
C = 200 - 5R
Where C is the number of cars rented and R is the rental rate.
To find the maximum income, we need to multiply the number of cars rented by the rental rate. The income (I) can be calculated as:
I = C * R
Substituting C from the first equation into the second equation, we get:
I = (200 - 5R) * R
I = 200R - 5R^2
To maximize this income function, we will take its derivative with respect to R and set it equal to zero:
dI/dR = 200 - 10R = 0
10R = 200
R = 20
So, at a rental rate of $20 per day, the agency should rent cars to produce maximum income.
Now let's calculate the maximum income by substituting this value back into our equation for I:
I = (200 - 5(20)) * (20)
I = (200 - 100) * (20)
I = 100 * 20
I = $2000
Therefore, when cars are rented at a rate of $20 per day, the car rental agency will generate a maximum income of $2000 per day.