 Imagine an open box is made by cutting a square from each corner of a 12 inch by 12 inch piece of metal and then folding up the sides. How much should be cut off so that the box volume is 128 cubic inches? So let's try to get some explanations going on here. So we start off with this metal sheet and the dimensions we are told in the first sentence here is that this is a metal square that's 12 inches by 12 inches. So if we if we look at that for example this dimension down here is going to be 12 inches long and then this dimension right here is going to be 12 inches as well. And so then it tells us that squares are cut off from each of the corners. So something like that. So we're cutting off these corners and so once the corners are cut off then we can use some type of you know shop tool to fold up these flaps and then maybe we weld them together or somehow we would then fold up these flaps so like this gets folded up this gets folded up and when you do that you're then going to create a box from the metal in which case you can then see on the bottom this square right here coincided with the original square then these flaps like the flap on the front and the flap on the back coincided with these right here and then the flaps on the sides this side whoops this side and this side coincided with these flaps right here. So we fold up the box and make something like this. Say it's an open box because it actually doesn't have a top there's only five sides to this to this box here one two three four and five so there's just the five that's that's what the word open is trying to describe in this situation great now so this is how we're going to make it so we have to decide how much do we cut off right how much should we cut off the box and so really what we're saying here is there's some unknown value x that we need to cut off and these all these little squares are going to be the same dimension here so how much should we cut off to form this box well we need the volume of the box to be 128 cubic inches that's a measurement of volume so if we think about the volume of a box right the fact that the top is missing really doesn't affect the volume whatsoever that would affect the surface area but it doesn't affect the volume so we need that the volume is going to equal 128 cubic inches but as this is a box we also know that we have length times width times height the volume should be their their product length times width times height now let's try to analyze what's going on here so we have to connect these variables l w and h to the variable we actually care about we need connected to this variable x how much do we cut off on the box now notice that as you fold this flap up the height of the box is actually going to equal this value x so h is actually equal to x that's great we're going to set up an equation in terms of our variable h now what about the length and width so something to point out here is the length of this box is going to be this dimension right here for which it is related to x right because if I take x plus l and then I add another x to that that's supposed to equal the whole dimension 12 so the thing is if I take away the x's that's how you get length and so the length is going to equal 12 minus 2x it's the whole length of 12 inches take away the two flaps of x that's also true for w right w is going to be this dimension right here and since this side was also 12 inches we get the length and width are actually equal to each other they're both 12 minus 2x and so that gives us a that gives us a position where then we can set up our equation we have that the height was x the length was 12 minus 2x and then the length the width was also 12 minus 2x this is equal to 120 128 cubic inches this polynomial equation is now what we need to solve and so notice that you have 12 minus 2x those are both divisible by two you can factor out a two and you can do that for the first factor and the second factor so notice I write this as 4x times 6 minus x squared is equal to 128 which sure we could divide both sides by four that's that's nice right here divide both sides by four and so that gives us x times 6 minus x squared is equal to 32 now to solve a polynomial equation we're probably going to want to set the right side equal to zero because factoring trying to fact factors that 32 is not as effective as fine factors of zero because there are a lot of different ways of factoring 32 many of which do not involve fractions or whole numbers so we really do need to set the right hand side equal to zero so we get x times 6 minus x squared minus 32 equals zero we need to expand this right here x minus a 6 minus x squared for which you could just foil it x minus 6 times x minus 6 you can also just use the binomial theorem which would tell us this is going to be 36 minus 2 times 6x plus x squared minus 32 equals zero then working that a little bit more we get 36 minus 12x plus x squared minus 32 I'm going to distribute the x now so we then are going to get an x cubed minus 12x squared plus 36x minus 32 equals zero so now this is the polynomial equation that we need to solve right here and so normally you know if I was approached with a polynomial equation like this I'd probably use a calculator or some some technology of some kind to solve this for me because solving polynomial functions can be very very difficult now the good news for us is that I did pick coefficients that we could solve purely algebraically using techniques we've seen before so if we look at the possible rational roots we need to look at factors of negative 32 this is going to give us 1 2 4 8 16 and 32 I'm ignoring negative possibilities because I can't cut off a negative amount on the I can't cut a negative amount off of my square so it's going to be a positive number so if there are negative roots don't care they're not going to solve this problem I need to have some positive ones uh never really like to try something too big so maybe something in the middle like four might be a good place to get started so if we try doing some synthetic division with that so right now my coefficients 1 negative 12 36 and negative 32 if we tried four whoops we bring down the one one times four is four minus 12 is negative eight times four is negative 32 like so plus 36 is positive four times that by four we're going to get 16 which is going to give me a negative 16 not quite there yet so four didn't work I'm actually going to erase these numbers four didn't work it wasn't necessarily too big we could try something else you know we could try something like maybe eight let's try that because again we didn't get we didn't get indicated that four is too big so if I'm playing this hot and cold game I'll try try eight here so you have one times eight is eight minus 12 is negative four times that by eight you're going to get negative 32 again which adds up to 36 to give you four and then four times eight is 32 which then you go then we found our route right there so our polynomial can factor as x minus eight that's one possibility and then we get x squared minus four x plus four equals zero so we need to find factors of four that add up to be negative four actually this is a perfect square trinomial this factors as x minus two quantity squared there equals zero so it looks turns out that we have potentially two feasible answers we have x equals two and we have x equals eight does the fact that x equals two the fact that it's a repeated route does that have much effect here really not so much we're just looking for the two numbers here two and eight okay the fact really got a square kind of had to do with the fact that the base of this box is going to be a square now which is which was the right answer two or eight does it doesn't matter maybe they're both answers now one thing you have to think about it would be sort of like the following situation right if x equals two if x equals two you're going to get 12 minus well let's actually plug them there you get 12 minus four right which would give you eight so the bottom would be eight the side would also be eight and then the height the top would be two there so we're talking about this eight by eight by two box which that does give us 128 cubic inches is the volume that's great what about eight if we tried eight you get 16 minus excuse me 12 minus 16 which is actually equal to negative four that's kind of weird uh how do you have how do you have negative four inches going on here this would also turn out to be negative four inches and then the thing is you could take negative four times negative four times eight that gives you 128 as well so the thing is the domain of this problem does not include eight so here this this answer the eight by eight by two that's coming about because x was two inches we want to cut off two from the square and so we have to we have a story problem it's very critically analyzed whether our answer even makes sense and that often comes down to the domain of the problem right here so I would argue that x equals eight is infeasible because after all how much can you actually cut off from these things so in some respect we ask the domain of this problem x has to be something non-negative you could potentially cut off nothing right um which case your volume would be zero it would just be flat right so really really you know we should probably say it's strictly greater than zero that's not going to give possibility uh if it goes again to cut off nothing but how big could you get it well like as x gets bigger and bigger bigger bigger bigger you're also getting bigger bigger bigger on the other side so at some point they're going to basically meet in the middle right and so basically there's going to be this point where you cut off four corners and then there's nothing left that would be halfway across so this would be x equals six and so the domain of this problem is going to be zero to six and that's the issue that happened here we found a solution eight to the polynomial equation that lives outside the domain of our story problem and so therefore we had to discard it which then tells us that x equals inch what what does that talk about x equals two inches that's what it was supposed to say x equals two inches is the answer we need to go with here we need to cut off two inches from the square to get a box that's 128 cubic inches