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.
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What you have done is completely correct.
You should state right from the beginning that
so you have
From the way the question is worded, it seems likely that the answer will be an integer
You could just plug in x=2,3,4 5 and see if one of them works
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
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.
[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.
If you were going to continue then you would use algebraic division for this.
The you would use the quadratic formula to find the other 2 solutions.