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The Energy Booster Company keeps their stock of Health Aid liquid in a tank that is a rectangular prism. Its sides measure $x-1$ centimeters, $x+3$ centimeters, and $x-2$ centimeters. Suppose they would like to bottle their Health Aid in $x-3$ containers of the same size. How much liquid in cubic centimeters will remain unbottled?

$\frac{12}{x-3} \mathrm{cm}^{3}$

Intro Stats / AP Statistics

Algebra

Chapter 12

Probability and Statistics

Section 6

Statistical Measures

Sampling and Data

The Normal Distribution

Introduction to Combinatorics and Probability

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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So 48 is kind of an interesting problem because it has multiple parts going on with it that are actually really good review. So we have or told we have a rectangular prism with various ah lengths with some heights the X minus one, the experts three and the X minus two. And so we're pretty much asked to find the violet This and I know it seems weird because there's a variable involved. There's an X, but were still able to multiply X minus one and expose three and X minus two using the foiling with it and then eventually we're gonna try and see how much we can fill bottles with if we from this material that's getting inside this, uh, prism. So we're going to go ahead and boil X minus one X plus three and explains to So we're going to foil the 1st 2 So foil means to do the first outer enter and then the last. When it comes to distribution, so x Times X is X squared, then X times threes. The X minus one x minus Read for the last one. So I'm gonna clean that up and then we switch colors X squared plus two X minus three and then this X minus two still needs to be multiplied. It comes down with that. Now I do something called the the box method, Uh, for when it comes to doing this kind of way with multiple with more than two. So I've got X squared going here to x here in the negative three goes here, my ex and you need to know what I'm trying to do is I'm trying to almost like a pocket square from science if you're familiar with that, but basically was gonna end up happening is I'm going. Teoh, um, meet and greet, so to speak was like meat and multiply instead. So this X and this expert will come together in this square right up in the up election corner, and that's gonna be X to the third, then the X squared. And then they going to do We're going to meet in this box, and that's going to be a native two x squared The X on the to exile meet in desert. That's gonna be two x squared to accept. Negative two is negative for X, my little three x and room negative two times three is a positive six. So what we look at and focus now is what could be combined. So I look at these. Looks like a little plus symbol here. Editors out negative to and to cancel out there you just cross it up. It's cancers now the negative four. Excellent. With three x days. Combine someone right in order my X cubed minus four plus 37 of seven X So my seven x and then plus the six. So this happens to be my volume in centimeters. Cute. Don't forget to label. So this is a volume. And if you're trying to figure out how many bottles that won't get used up, if each bottle has a volume of X minus three, we're going to just do some division. So step to is going to be to do volume of the prism divided by the volume of the bottle. And it's best to use what we call synthetic division. So that's called since that it division cation you to review that. And what I basically do is I write out they the coefficients, um, the numbers in front. So the cone of the volume So there's there's ah ah one here but the one here. Negative seven at six. So I draw myself a box and I got a one negative seven and a six. But I have to be careful right here. There's an extra third and his ex. So in reality, I need to make into account that there was technically a zero here as well. And it goes in order from highest degree down the lowest degree. And so here's really have to be very careful. If the volume is X minus three, then that means the number in front right here, usually a positive re. So I change the sign. That's very important. So we're ready for some synthetic division now. So the first thing you do is a drop. Found the one and then I multiply through three times. Positive. Three times one is three and light here. Zero post three is three. Three times three is nine nine plus nine. You have seven is too. And then we've got two times three IHS six, remember? Ah, we're adding here. Every number I get here means that's my remainder. So the question was asking, What is your remainder of your volume when you feel the bottles and it's always going to be the number that you get in that box over what you were dividing vice in this case X minus three. So that would be, um, Bottles. It'll get Armfield.

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