how much is charged to maximize revenue. calculus problem?
A certain toll road averages 36,000 cars per day when charging $1 per car. A survey concludes that increasing the toll will result in 300 fewer cars for each cent of increase. What toll should be charged in order to maximize the revenue?
Let x be the number of cents we increase the toll.
So an expression for the revenue would be:
(36000 – 300x)(1+.01x)
When we FOIL that expression, we get
36000 + 360x – 300x – 3x^2
-3x^2 + 60x + 36000
So the revenue function looks like this:
f(x) = -3x^2 + 60x + 36000
To find the maximum point, we can take the derivative of this function and find when it equals zero:
f’(x) = -6x + 60
-6x + 60 = 0
-6x = -60
x = 10
So the toll should be increased 10 cents to $1.10.
Lec 28 | MIT 18.01 Single Variable Calculus, Fall 2007