Calculus Problem Of The Day

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

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6 Responses
  1. LM says:

    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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  5. Andrew says:

    It doesn’t require Calculus but is alot faster with it.

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