SOLUTION: A new tollway is built. For the first month there is no toll to use the road and each day 4800 cars use it. After the first month it is estimated that for each dollar increase in t

Question 894328: A new tollway is built. For the first month there is no toll to use the road and each day 4800 cars use it. After the first month it is estimated that for each dollar increase in the toll, 800 fewer cars will use it. What price should the company set the tolls in order to make the most money?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
demand = 4800 when the price is 0.
demand = 800 less for each dollar increase in the toll.

revenue = price per car * number of cars.

your demand goes down 800 for every 1 increase in price per car.

the formula for demand would be numbers of cars = 4800 - 800 * price per car.

let d = number of cars
let p = price per car.

the formula becomes:

d = 4800 - 800p

when p is 0, the demand is 4800.
when p is 1, the demand is 4800 - 1*800
when p is 2, the demand is 4800 - 2*800
etc.

your revenue is equal to the number of cars * price per car.

let r = revenue.
let d = number of cars
let p = price per car.

your formula becomes:

r = d * p

since d = 4800 - 800p, you can replace d in the equation with that to get:

r = (4800 - 800p) * p

simplify this to get:

r = 4800p - 800p^2

rearrange the terms to get r = -800p^2 + 4800p

this is a quadratic equation that can be solved for the maximum value.

replace r with y and p with x to get:

y = -800x^2 + 4800x

set y = 0 to get:

-800x^2 + 4800x = 0

maximum value for this equation is when x = -b/2a

since this equation is in standard form of ax^2 + bx + c = 0, you get:

a = -800
b = 4800

x = -b/2a becomes x = -4800 / -1600 which becomes x = 3.

when x = 3, -800x^2 + 4800x becomes -800(3)^2 + 4800(3) which becomes -800*9 + 4800(3) which becomes -7200 + 14400 which becomes 7200.

the maximum revenue occurs when the price per ticket is 3 and the maximum revenue then becomes 7200 dollars.

here's a graph of your equation of y = -800x^2 + 4800x.

$$$

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