Question

when a school play charges $2 for admission, an average of 100 people attend. For each 10 cent increase in admission price, the average number decreases by 1. what charge would make the most money?

Answer 1

$6

Explanation:

We know that when the price increases the attendance will be lower compared to that. Let’s create model and determine the function for this:

if ,

99 people = 99 x $2.10 = $207.90

98 people = 98 x $2.20 = $215.60

97 people = 97 x $2.30 = 223.10

96 people= 96* $2.40= 230.4

95 people= 95* $2.50=237.7 and etc. so we see it is going on rising trend but there will be break even point.

The function or model for the revenue will be:

f(x) = (100-x) * (2 + 0.10x)= 200 - 2x + 10x - 0.1x^2

If we know the function is this: F(x)= 200 + 8x - 0.1x^2

Let’s differentiate it to be sure what is the best price regulation to make more profit

Df(x)/dx = 8 - 0.2x

No we need to find max set to 0 and it will be:

8-0.2x = 0 and,

x = 40

So now we are able to know 60 is the best average of people, which would say to us that th admission price is 2 + 0.10*40 = $6