Marge is planning a casino trip.If 100 people sign up, the cost is $300 per person. For each add. person above 100, the cost per person is reduced by $2 per person. To max. Marge's revenue, how many people should go on the trip? What is the cost per person?
(Calculus)

Question

Marge is planning a casino trip.If 100 people sign up, the cost is $300 per person. For each add. person above 100, the cost per person is reduced by $2 per person. To max. Marge's revenue, how many people should go on the trip? What is the cost per person?
(Calculus)

anonymous · Answer

PLEASE HELP!

valpey · Answer

If 250 people sign up, what is the cost per person?

valpey · Answer

No. Marge's revenue is equal to the number of people times the cost per person. We know that when there are 100 people the cost per person is $300. The revenue at 100 people is 100*$300=$30,000. Let x be number of people, total revenue be r(x), and cost per person be c(x) then:
$$r(x) = x*c(x)$$

valpey · Answer

But c(x) goes down by 2 for each additional person (above 100) so 
$$c^{\prime}(x) = -$2\text{ per additional person}$$

valpey · Answer

Therefore we can say:
$$c(x) = -$2*x+C$$
for some constant C. What does C need to be for c(100) = $300?

valpey · Answer

No.
$$c(x)=$500-$2x$$
$$r(x)=x($500-$2x)=$500x-$2x^2$$
$$r^{\prime}(x)=$500-$4x$$
If we want the local max, we should set r' equal to 0 or:
$$$500=$4x$$

valpey · Answer

The number of people should be above 100 and below 250.

anonymous · Answer

I figured it out it is 125 people at $250 per person

valpey · Answer

Sweet. Well done.

anonymous · Answer

is this a calculus question?

valpey · Answer

The asker wrote "(Calculus)" but obviously, we don't have to think of it that way.