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.

 Nov 4, 2020

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


You could just plug in x=2,3,4 5  and see if one of them works frown




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.


\(\pm1,\;\pm3,\; \pm81\)

[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.



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.



 Nov 4, 2020
edited by Melody  Nov 4, 2020

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