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## Homework Statement

A cruise ship that can hold up to 62 people provides excursions to groups of 42 or more.

If the group contains 42 people, each person pays $70.

The cost per person for all members of the party is reduced by $1 for each person in excess of 42.

Find the size of the group that maximizes income for the owners of the ship.

## Homework Equations

## The Attempt at a Solution

(i) Income = (ticket cost)(number of people)

Let x = the number of people and ##p\left(x\right)=70-1x## is the cost per ticket.

Then,

##I(x)=x⋅p\left(x\right)##

(ii) ##I\left(x\right)=x\left(70-1x\right)##

##=-x^2+70x##

##=-\left(x^2-70x+1225\right)+1225##

##=-\left(x-35\right)^2+1225##

(iii) Maximum income is $1225 when the number of people is 35.

Now this answer doesn't make sense, as the minimum amount of people required is 42.

I don't know how to incorporate the range limits of people into the equation. That is, there are only 62 people allowed on the ship, and there is a minimum requirement of 42 people.

Please bare with me. I've been posting a lot of questions, but this is for online classes and there is no lecture to ask questions!