Question 894328
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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