So, I'm trying to help someone with some highschool math, and i've got a question that's just gone over my head. Could someone give me a hand with this?

Ten-year-old Kayla's lemonade stand has experienced rises and falls in sales caused by temperature changes over the last few summers. Her sales over the first two weeks of summer were tracked according to the model:

S(x)= x^3 - 12x^2 + 36x

Where "x" is the number of days and "S(x)" is the number of sales. She makes a profit when she sells at least 10 glasses of lemonade. Determine the number of days that she made a profit.

Much appreciated.

-Fooglmog
Guy with no clue.

... 11?

-Fooglmog
Guy with no clue.

The easiest way to do it (but won't work in general) is to factor the function. I don't want to solve a cubic equation.
note that we know that x>0 fo rthis problem
x^3-12x^2+36x = 0
x(x-6)^2 = 0
x = 0, x=6
also I know it'a cubic equation so s(x) will either be increasing or decreasing after 6. So lets plug in numbers

x S(x)
1 25
2 32
3 27
4 16
5 5
6 0
7 7
8 32
So the answer is on days 1,2,3,4 and after day 7.
so 1,2,3,4,8-14 which is 11 days.
This method won't work with any equation but for this problem it's the fastest way.

graph it and draw a line?

http://www.wolframalpha.com/...+%3D+x^3+-+12x^2+%2B+36x+

x^3-12x^2+36x = 10

I was going to factor it in this form, but wolfram alpha informs me that that would be hellishly complicated

in order to solve this type of thing in general it would get complex.
http://en.wikipedia.org/wiki/Cubic_function. The equations for solving this are in there

You can also solve quartic functions exactly too. :P

Alternatively you can use newton-raphson to solve the inequaltiy x^3 - 12x^2 +36X > 10
subject to x>0 and x<15 but this is rather overkill.

urgh, I just noticed that I forgot a lot of the math I learned at university, lol

as an aside you can change any polynomial optimization problem into a linear optimization problem, which mathematically makes it easier, but they solve in polynomial time. Quadratic optimization problems can be tweaked to run faster.
That's all I remember from grad school on the subject:P

Alternatively you can use newton-raphson to solve the inequaltiy x^3 - 12x^2 +36X > 10
subject to x>0 and x<15 but this is rather overkill.

Ooh! Ooh! Do it do it do it!

nerds

x^3 - 12x^2 +36X >= 10

"She makes a profit when she sells at least 10 glasses"

That's all. :p