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Using Models Use the model given to answer the questions about the object or process being modeled.

The portion of a floating iceberg that is below the water sur-face is much larger than the portion above the surface. Thetotal volume $V$ of an iceberg is modeled by$$V=9.5 S$$where $S$ is the volume showing above the surface.(a) Find the total volume of an iceberg if the volume show-ing above the surface is 4 $\mathrm{km}^{3}$ .(b) Find the volume showing above the surface for an ice-berg with total volume 19 $\mathrm{km}^{3}$ .

a)$\begin{aligned} V &=9.5 S \\ V &=9.5(4) \\ V &=38 \mathrm{km}^{3} \end{aligned}$b)$\begin{aligned} V &=9.5 S \\ 19 &=9.5 S \\ S &=2 \mathrm{km}^{3} \end{aligned}$

Algebra

Chapter 0

Prerequisites

Section 1

Modeling the Real World with Algebra

Equations and Inequalities

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this question describes the mathematical relationship between the amount of ice above of an iceberg and the total amount of ice in the iceberg. Now, for questions like this, where you have some sort of, uh, word problem, your job is to translate that to some math equation and then take that math equation and plug out on answer to whatever questions that they give you. This is this is the step, usually where most people, including myself, have problems. For a lot of questions like these, the actual calculations that you're gonna do, we're gonna be pretty straightforward. But it's taking the information in context and distilling it somehow down the actual math that weaken. D'oh! So answer this question. We're gonna take a little bit of time and make sure that we understand exactly what we're given and what we're being asked to calculate. So there's two variables in the equation. This equation, the first is V, which is the total ice in the iceberg, and the units that they give us for that one are Kilometers Cube. It's important to note that if they give you variables with units in them than you answer that you send out also has to be in units. So this is something that we're gonna be keeping track of throughout the problem. The second variable that they give us is s, which is ice above the surface of the water. I'm not gonna write all of this out because it's not overly important. This is just to remind us exactly what we have and how we're gonna deal with it. Now, the equation that actually makes this question interesting that relates Thies too, is that V is equal to 9.5 s and the structure of the questions that were asked Is there gonna give us one piece of information about this equation and we're just gonna have to find the other one. Writing at all of the information like this is helpful for a couple of reasons. One, it really makes you understand what everything is. And also now I don't have to look back at the big, long paragraph or two. If I if I forget what s is or what the equation is. I just have this nice little cheat for myself, and at least for me, I'm far less likely to make mistakes with this amount of information that I am with two paragraphs worth of information. So for the part A of the question we are given explicitly, that s is four kilometers cubed and they want us to calculate what V is and the way we're going to do this is this equation. We have an equation with two variables in it, and we know one of the variables, so we can always calculate what the other one is. It's the first step to this is just too pretty naively, just right out with the equation is, And whenever you're trying to solve for a variable, you want to make sure it's by itself on one side of the equal sign that we can plug in everything else ever hear calculated out. And then we get the answer that we want. So for here, it's just a pretty simple substitution because we have that s is exactly four kilometers cube. When everybody see an s, we could just replace it with four kilometers cubed and go through and calculate what the actual answer is. Units are multiplied in the same way that variables and numbers are. So actually, we're gonna group this. We're gonna group These two together and multiply them and we have the nine times four is 38 then we tack on the units at the end. We're just kilometers cubed now a good sanity check, which was great because we did all this preparation of here is this you do our units for V and the answer that were claiming matched the same that we know it has to be from the problem. And in this case, yes, we have kilometers cubed here and kilometers cubed here. This was cloudier, squared or feet per second or interests per pound or any kind of crazy unit. You would instantly know that there's no way we got the right answer Now we're a little bit more confident that this is correct. Given that we have the right units, it's important to go through and check to make sure that your math is still correct. But we're a little bit sure that our process, at the very least, is is right. This is a good habit to get into, um maybe not for the simpler problems. But whenever you're dealing with a lot more complicated stuff, so you might have five or six variables and four or five equations. Doing this kind of prep work is gonna be incredibly important. And also, if you're doing this on the test, your quiz and it's really used to go back and check your work and see OK had these variables. And I did this process and I got this answer and you might be able to see ah, something that you did wrong very quickly. So for part B of this question is just the reverse of the first were given information about V, which is that it's 19 kilometers cubed and they want us to calculate what s is this is about this is no more difficult than the previous one. It just requires an extra step. Because when we write out our equation, we see that the thing that we want to solve for S is not by itself. So we have to isolate it. So the thing that's being it's being multiplied by as 9.5. So we want to divide both sides by 9.5. We have the same thing that's on the top of the bottom here. So these cross out and we're just left with s by itself. So I'm going to rewrite that as just s is vey over 9.5 and then we can plug in what we value that we were given for me. So when we plug in 19 kilometers cubed over 9.5, we get that s is two kilometers cubed running out of space there at the end. So again, quick unit check dis kilometers cubed makes sense for s Yet because we wrote it right here And another important thing to note when you're doing these problems is it's a usually a good idea to save plug in the numbers for the end. You could just immediately take this equation and plug in 19 kilometers cubed here. But at least I am far more likely to make mistakes with no numbers than variables. And there's track of here, so it's usually a good problem, A good habit to get into, to manipulate all of your variables first, and then plug everything in usually makes it a lot easier to check your work. And if someone else is trying to figure out what it is, it is just a lot cleaner to read hope that was helpful

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