# Calculus Problem Of The Day

**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**

Calculus isn’t required for this problem; there are two roots of f(x) = -3x^2 + 60x + 36000, 120 and -100

so you find the average of the roots to get the abscissa of the vertex: x = 10. This is logical by the symmetry of parabolas.

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