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# [Grade 12 Pre-Calculus: Polynomial Functions]

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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

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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.



Nov 4, 2020
edited by Melody  Nov 4, 2020