Answer:
It take 65 people to maximize the revenue.
Step-by-step explanation:
Consider the provided information.
Let x is the number of people who take part in trip.
They charge $50 per person and they will reduce the price per person for all riders by $0.50 for each person excess of 30.
The price for each person is [tex]50-0.50(x-30)[/tex] where x is greater or equal to 30.
[tex]50-0.50x+15[/tex]
[tex]-0.50x+65[/tex]
Now write a revenue function by multiplying the number of people with the price per person.
[tex]f(x)=x(-0.50x+65)[/tex]
[tex]f(x)=-0.50x^2+65x[/tex]
We need to maximize the company's revenue.
The above function is a parabola that opens downward because the coefficient of x²is negative.
Therefore, the maximum of the function is at its vertex.
If the equation of the parabola is [tex]f(x)=ax^2+bx+c[/tex] then we can find the coordinate of vertex at [tex](\frac{-b}{2a},\frac{4ac-b^2}{4a})[/tex]
Calculate the value of [tex]\frac{-b}{2a}[/tex] for the function [tex]f(x)=-0.50x^2+65x[/tex].
[tex]\frac{-65}{2(-0.50)}=65[/tex]
Therefore, it take 65 people to maximize the revenue.
