Name: Craig
Who is asking: Parent
Level: Secondary
Question:
A dance studio charges $80 per student for a series of 2 hour lessons.
The studio's costs are $30 per hour for the instructor, $15 per lesson
for the room rental, and $3 per student for miscellaneous expenses. If
x is the number of students enrolled in the class, express the studio's
profit P(x) in terms of x. Find the profit if 10 students are enrolled
in the class.
I would like to know how to set this up so that I can help my son.


Hi Craig,
The dance studio charges each student $80 for a 2 hour lesson, but it
doesn't get to keep the entire $80, it has some expenses. The profit is
the income (what the students pay) minus the expenses. There are three
expenses.
 $30 per hour for the instructor
 $15 per lesson for the room rental
 $3 per student for miscellaneous expenses
If there are x students then the income is $80 per student
so the total income is $80 x.
The expenses are:
 The way I read this is that the instructor charges
$30 per hour, regardless of the number of students. Since this is a
2 hour lesson the instructor charges $60.
 Rental of the room for 2 hours costs $15.
 The miscellaneous expenses are $3 per student hence,
if there are x students, then the total miscellaneous expenses are $3 x.
Since the profit, P(x), is (income)  (expenses)
P(x) = $80 x
 ($60 + $15 + $3 x)
Cheers,
Penny
Thank you for the help, but I have one further question. My son has
a formula that goes like this: Profit = x(mnx)(a+bx) with x =
the number is students
a= fixed costs
bx= variable costs
the above I see in the answer, but I am not sure about the mnx.
He says it has something to do with supply and demand.
I can see where the equation you posted is good for the profit, but is
there anything more you can tell us regarding the above information.
Thank you again.
Craig
Hi Craig,
It really bothers me to see mathematics presented as a collection of
formulas.
The m  nx might represent supply and demand, or some kind
of discount. Maybe, to try to attract more students, the dance studio
trys to get customers to encourage their friends to join with the following
fee structure. The basic cost per student is $80 but every time another
student joins, every student gets a $2 discount. Thus if there is
only 1 student the cost is $80. If there are two students then each pays
$78, if there are three then each pays $76,... With this plan each
student pays $82  $2 x,
where x is the number of students. Hence the profit would be P(x)
= x ($82  $2 x)
 ($60 + $15 + $3 x)
Penny 
