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Kennesaw State University

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Problem 99

A roofing contractor is fabricating gutters from 12-inch aluminum sheeting. The contractor plans to use an aluminum siding folding press to create the gutter by creasing equal lengths for the sidewalls (see figure).

(a) Let $ x $ represent the height of the sidewall of the gutter. Write a function $ A $ that represents the cross-sectional area of the gutter.

(b) The length of the aluminum sheeting is 16 feet.Write a function $ V $ that represents the volume of one run of gutter in terms of $ x $.

(c) Determine the domain of the function in part (b).

(d) Use a graphing utility to create a table that shows sidewall heights $ x $ and the corresponding volumes $ V $.Use the table to estimate the dimensions that will produce a maximum volume.

(e) Use a graphing utility to graph $ V $. Use the graph to estimate the value of $ x $ for which $ V(x) $ is a maximum. Compare your result with that of part (d).

(f) Would the value of $ x $ change if the aluminum sheeting were of different lengths? Explain.

Answer

a) $12 x-2 x^{2}$

b) $2304 x-384 x^{2}$

c) $0 < x < 6$

d) 3$i n \times 6 i n \times 192 i n$

See table

e) See graph

f) No

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## Discussion

## Video Transcript

section complex Ones like this where the It's folding a piece of sheet metal. Like the edges up that are each length X, um, where it was like a foot wide. And come it folded up. Act someone side Excellent the other. So the bottom is 12 minutes to acts and the sides are gonna be X. Yes. Of a part name? Yes, for, uh, the height of the sidewalls X and read a function A That represents the cross sectional area case we're gonna have function A of eggs equals Well, the cross sectional area would be just area of this rectangle. The link times with for the speed 12 X minus two X squared. Okay. Part being says the length is 16 feet and write a function V represents the volume in terms of facts. Okay, so the envy of axe. Okay, So 16 feet is different from inches. We have toe convert that first eso 12 times 16. There's 192 somebody small to find by the length of the gutter. So 192 x Um oh, no. Okay, so 12 times 16 is 1 92 That's the length and inches. So we still have to multiply that by 12. So 92 times 12 is Ah, 2000 304 X minus 1 92 times two. I think that's 3 84 Let me just make sure 380 floor X squared. Eso that volume will be in cubic inches. Okay, perfect on apart. See, that says, determine the domain. So the domain like we can We can't let the smallest we can fold up zero But even then, we probably wouldn't do that s o for part c just real quick. Our domain the least We could fold a zero, but then it wouldn't be a gutter. So let me do an interval notation for this started zero with a parentheses because it it won't equal zero. And it can go until X is equal to six. Because then it's just folding a sheet piece of metal in half, and it's no longer a gutter, so it can't equal six. But it could be everything up to six and still be kind of considered a gutter. Okay, sir, demand goes from 0 to 6 for D. So there's a graphing utility to create a table that shows Sidewall Heights X and corresponding volumes of the Okay, So let's do that on a new slide. So part Dean, just gonna remember formula. So vivax equals 23 04 acts minus 384 x squared. So using graphing a tiptoe utility to create a table of values But at my ex be here and be here. So if x zero r volume is zero and we can go 123 45 and six came at six are volume will also be zero because then it would be folded flat Case is somewhere in the middle will be the maximum. Ah, so if we plug in, let's just go over to my calculator, Okay? So if f it's one, uh, the volume will be 1920 if X is one. Sorry, FX is too. Our volume is gonna go to 3072. If X is three. Uh, we're at 34 56. Starting to slow down to exit four is back to 30. 72 and five will be back to 1920 because it's symmetrical because it does Think of Parappa. So I'm guessing that three is my maximum amount to me. My maximum volume. Okay, so Burki says graph f and in he's the graph to estimate. Okay, So because we have to change our window, that's our X axis. Only needs to go from Sade a speak it Negative line. That's a seven. Then our Y axis. It can probably go a lot higher, since we would expect it to be over 4000. Let's make it go from negative 100. So you can still see underneath the X axis to 4000 because that's a good graph. Okay? And if we, um So I guess it's just ah probably gonna look something like this, and the maximum is gonna be at three. So using the graphing utility also confirms that our maximum will be at where we cut where we fold it, where X is equal to three. Um, so that should be about the same in part F. It says if the length changes, will it will it change what our answer would be? And I would say no, because the only thing that can really change with X is the cross sectional area. And so the length of the gutter, uh, as long as me, because if the length changes and the whole volume will be increased by that amount, But what's gonna affect the entire volume the most when our formulas in terms of X is just gonna be the cross sectional area? So changing the length I won't change our answer of folding at X equal story. Okay, so X equals three is the place to fold to get the most cross sectional area for a gutter. Okay, thank you very much.

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