A hall charges $30 per person for a sports banquet when 120 people attend.
For every 10 extra people that attend, the hall will decrease the price by $1.50 per person. What number of people will maximize the revenue for the hall?
Let x be the number of people attends.
Let the revenue be y.
Let the initial revenue when 120 people attend is 30*120 = 3600
y= people attend * price
y= (120+x)*(30- 1.5*(x/10)]
y= (120+x) * ( 30 - 0.15x)
y= ( 3600 - 18x +30x - 0.15x^2)
==> y= -0.15x^2 +12x +3600
Now we need to find the maximum values.
First we will determine y'.
==> y' = -0.3x +12
Now we will find the critical values.
==> -0.3x +12 = 0
==> -0.3x = -12
==> x = -12/-0.3 = 40
The number of people that will maximize the revenue is 40 plus the initial 120 people = 40+120 = 160 people