+16 An open box is made from a piece of rectangular cardboard of 12cm by 15 cm, by cutting square corners, x cm by x cm, and then folding up the sides. What size of each square must be cut to have a volume of 162 cm^3, where x is greater or equal to 2?
So far I got 2x(x-6)(2x-15)=162cm^2 but I'm not sure where to go from here.
+118667 Hi May,
It is nice to meet you.
What you have done is completely correct.
You should state right from the beginning that
\(2\le x<6\)
so you have
\(2x(x-6)(2x-15)=162\\ \)
From the way the question is worded, it seems likely that the answer will be an integer
so
You could just plug in x=2,3,4 5 and see if one of them works 
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However, if none of those work, or if you want to do it in a more complicated way then here is an alternative.
Expand it and take it all to the same side. (start by dividing both sides by 2)
You will get
\(2x^3-27x^2+90x-81=0\)
You need to look for factors and the way the question is worded it seems that there will most likely be an integer value of x
If there is an integer one to find it will be a factor of 81.
choices
\(\pm1,\;\pm3,\; \pm81\)
[edit: I forgot +/-9]
I usually look at the plus and minus at the same time as it saved time, but I am hoping for a pos one so not this time
Normally Id start with +/-1 but we are looking for one between 2 and 6 so I will try 3
If x=3 then
LHS=2*27-27*9+90*3-81 = 0 = RHS
So x=3 will make this work.
You can continue on to look for other possibilities if you want but you already have your answer.
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If you were going to continue then you would use algebraic division for this.
\((2x^3-27x^2+90x-81)\div (x-3)\)
The you would use the quadratic formula to find the other 2 solutions.
\(\)
