A caterer provides a conference banquet for a price of $40 per person for 20 or fewer people but will decrease the price by $1 per person for everyone if there are more than 20 people. What number of people will produce the maximum revenue for the caterer?

The maximum revenue is $900, obtained with 30 people

Step-by-step explanation:

Naturally, the answer should be a number equal or higher than 20, because up to 20 persons, each one pays the same. Lets define a revenue function for x greater than or equal to 20.

f (x) = x * (40 - (x-20)) = - x²+60x

Note that f multiplies the number of persons by how much would they pay (here, assuming that there are more than 20).

f is quadratic with negative main coefficient and its maximum value will be reached at the vertex.

The value of the x coordinate of the vertex is - b/2a = - 60/-2 = 30

for x = 30, f (x) = 30 * (40 - (30-20)) = 30*30=900

So the maximum revenue is $900.