sketch-the-indicated-curves-by-the-methods-of-this-section-you-may-check-the-graphs-by-using-a-ca-11

Question

Sketch the indicated curves by the methods of this section. You may check the graphs by using a calculator.
A rectangular box is made from a piece of cardboard 8 in. by 12 in. by cutting equal squares from each corner and bending up the sides. See Fig. $24.45 .$ Express the volume of the box as a function of the side of the square that is cut out and then sketch the curve of the resulting equation.

Dorcas Attuabea Addo · Accepted Answer

right. So we are looking at ways still trying to find the volume. Uh ah. Reaching Bounded by way to be false. So one over its way. A way to be quotes uncertain. So we have the eggs equal to sue a eggs to be pulled towards it. You realize that if he&#x27;s catch states, you have So let&#x27;s see, I have I see this is my eggs and this is words way this case. So let&#x27;s see. This is when I have one year his sketch you grab in your graph. It&#x27;s looking my day this is sounding would look like for why should be put to wear number one x Great. It&#x27;s big. We would would like to find the volume off this reaching so to find there for you We had seen legs be which is the balloon be call today You see a to be the interval sue Bye f off X year minus I g off eggs the way eggs don&#x27;t forgives you have f of X gates. That and g off woods a both you. So let&#x27;s eso already. We know this Geo X. It&#x27;s what zero Do you realize that I have my Secret Day interval. It&#x27;s from words soon, so they eat so well from soon from 2 to 8 to buy out. So I have X and f of X is going to be words one by Woods. Thanks. Would Wade the okay? This is going to give me soup by you. See growth soon to AIDS off words one. Overweight eggs The woods be eggs. Why one of my eggs? You realize that you have X year and eggs Great here. So these AIDS consuls that&#x27;s also have won over woods A. From here, you realize that if I find a sea grass off this, what do I have? I have to Hi lane whites. Hey, from suits Who would eight so to buy. Let&#x27;s from the boundaries lane. It&#x27;s minus words lane words Sue Don&#x27;s forgets time Express This us was. I can write this as soup. I name late AIDS. It&#x27;s what&#x27;s so cute? No, but I waas soon to keep my words too. This is one. Don&#x27;t forget. You can still express your and laying in this words in this form. So here you realize that what do I have? Do cancels they So you realize I&#x27;m going to have to buy lane off Woods. Su would squeeze and you kind of right. So if I have a su lane a Don&#x27;t forget, I can write this us words. Plane was eggs scrape. So my suit kind. Still calm. So they have words soon off Woods Dubai Lane soon so that this gives me one. Well, hi, lane. Words soon. So this is Woz. My volume. And because it is value they like I rights Waas Units, Words que the units there for you. I am sorry. So this it&#x27;s going to be one our final result.

Matthew Lee · Answer

we&#x27;re going to begin by completing a table that they give us. So, uh, com link and with which are equal because we&#x27;re starting with square image problem. Uh, this column, rather. And then the volume in the final collars. And note that volume before too lame on with champagne. You already given birth to thank you for the third column one hiding three the name Inuit 24. Mining three on each eye. So your time three. You&#x27;re the volume if you want you the height three I&#x27;m going. And the wind, which are the same. So it will be 24. My 203 squared. Do you know it would get you 972 and I&#x27;m going, Going the other? Uh yeah, the road. So the next one going the next day, which would be 24 minus two times before you could get some calculation for the volume you&#x27;ll get when docking 200. I run out in 24. Bye. No high tech spy tracks. Five. Twice on divine you&#x27;ll get will be. I had a date and I&#x27;m just going to write to volume for them last. I have six. It is going to be 854. All right, uh, next we&#x27;re going Teoh, Right before move our basic just the table would look like the maximum height is somewhere between, uh, four or five. Somewhere around here, your own for um, So on the next part, we&#x27;re going to write volume of the function of X or ex is the height. Ever call that blind? My time Time fight in recalling Hey, X, Thank you. Wit are both 24. You look at the picture you could track back to me. Each side there were using Theo morning 24 months back, and then the weight is off 20 former director and then the height of that. All right. And I just wanted by everything out on after expanding this and writing it, uh, and you can order we&#x27;re going to get for X Cube two. Excellent. Jackson Dex. I mean, minus 96. Like a square. Uh, right, 5 76 part B heart. He said calcalist to find a volume, which means we need to find the minimum of find a critical number, that finding critical number. We have a function of X. Now that we can take. The derivative derivative of the function is 12 x squared, minus 192 556. On the critical number, we said it&#x27;s derivative equal to zero. So you were a hero and this quadratic working doctor? 12. Actually, when the doctor out of 12 will get X squared, my neck I want to create and I hit a quadratic that can be factored. That would do on enough age. We can often divide inside the question by 12 to get rid of that 12 there. So we&#x27;re going to have zero equal. Factoring that quadratic, it&#x27;s not good. X minus 12 and X minus four. They&#x27;re Jew critical number are actually Quatro. And it was actually four. Uh, and remember previously for a call from our table that four appeared to be the maximum. And based on our find Crippen Emery, we found that the four is a maximum. Um, we could test further. You have the for a derivative? Well, you&#x27;ve already seen from our table is going to be the maximum. But if you wanted, we could also have some point in here. You&#x27;re a hero. You&#x27;ll find the derivative and positive. They&#x27;re gonna find you find the derivatives negative. And then I&#x27;m gonna say 20 We&#x27;ll find a derivative is positive. Uh oh. This number for a crew for the Mac is in action. They got your contrition dp here. Uh, furthermore from the problem, we know that ex can&#x27;t even be 12. Because I&#x27;m originally we were measuring height Megan with and at the height of 12 that makes a Lincoln with 24 minus 24 which would be a language bureau. So excuse me, I&#x27;m not about to be 12. This is. Or if it was, you&#x27;d get a bottom of euro. Um, for the final step, it has to, uh, you can grab him. Utility utility. Such a demo. You&#x27;re fine. Graphic. Something like this. Um, for your jing goes up and then come back down and have another architects that 12 this point up shared a maximum before the ground kind of graph, You will get you grab on a graphing calculator

Taylor Shimono · Answer

So for this problem, we want to find the surface area of the volume that is going to be created by writing wth e equation y equals a squared X about the X axis. So that means that we are going to use our formula. But she tells us, as the surface area is equal to the integral from a to B of two pi ethnics time to square root of one plus f prime of X squared DX I&#x27;m so we are explicitly given our A and B in the problem. So all we need to do is figure out our FX and Ephraim Lex. Um and our equation that were given is already in the form of FX, which is just the square root of X. So we know that Ephraim X is equal to one over to Room X, and I&#x27;m actually going to go ahead and square this before we plug it in to make our calculations with inner integral a little bit more simple. I&#x27;m so f of X squared is just one over for X, and now we have all of our missing information for calculating our surface area. So this will be equal to the integral from 2 to 6 of two pi times f of X just squirted X times the square root of one plus one over for X dx. And right now this looks pretty complicated. But we can actually go ahead and rewrite the inside of our radical, um, as the square root of for X plus one over for X. And so this denominator of the square root of four X is going to be two x, which actually cancels out with this to root ex on the outside of our radical. So we&#x27;re just left with the integral from 2 to 6 of protest square root of four X plus one DX. And we know that we&#x27;re going to be able to evaluate this using U substitution. So I&#x27;m going to set you equal to four X plus one. That means that d U. Is going to be equal to four d x. So you plug this into our internal. This will give us priority for times integral from 2 to 6 of you to the 1/2 power. Do you and so simplifying further when we integrate, we know that we&#x27;re going to multiply by the reciprocal of our ex wanted plus one. So 1/2 plus one is three hugs. So we&#x27;re going to multiply by 2/3 the reciprocal sometimes you, which was for X Plus one raised to the power of the exponent plus one. So, to the three hours hard, he validated from 2 to 6. And when we apply the fundamental serum of calculus and simplify, um, we are going to get that are, uh, surface area of this volume is going to be equal to 49 price over three square units.

Taylor Shimono · Answer

So for this problem, we are going to be checking the surface area that results when we rotate like ALS expert about the y axis. So that means that the formula for a surface area that we&#x27;re going to be using is the integral from CD of two pi times G a y times the square root of one plus g prime of life squared D Y I&#x27;m So all the information that we&#x27;re given in this problem is in terms of X. So that means that we&#x27;re first going to want to find, um g of why so all we need to do is take the square root of both sides of our equation when we get g of wise equal to the square root of why. So when we calculate g prime of life, it&#x27;s going to be equal to one over to root wife. And to make this easier to substitute into our equation, we&#x27;ll just go ahead and find g prime y squared right now which is equal to one over four. Why? Oh, and the next you want to find our boundaries in terms of why so the point X equals zero is going to correspond to y equals zero and the point X equals two is going to correspond to y is equal to four. So those will be our C and D values. And now we have all of our information to calculate the service area, which is going to be equal to the interval from 0 to 4 of two pi times. Ju Why would, she said, with square to fly times the square root of one plus g ive y squared, which is one over four y do I. And so from here we can go ahead and combined the terms in our second radical into one fraction, which is going to allow us to cancel out some terms and make this a little bit easier to integrate. So this is the integral from a 0 to 4 of two pi 10 squared of white of the square root of four wife plus one over four. Why do I? And so we know that the square root of four why is the same thing as writing to route? Why so our denominator is just going to cancel out, and this will give us the integral from 0 to 4 of pi times the square root of four wife plus one, Do I? And we could go ahead and do a U substitution. Here we&#x27;ll go and set you equal to four worthless one. So that means I do. You is equal to 40 y And so when we plug this back into our interval, our surface area is going to be the integral from 0 to 4 of a pi over four, which will go and bring that coefficient to the outside of you to the 1/2 hour. Do you? So we integrate. This is pi over four times using our powerful, sometimes two or three terms you which was four y plus one to the three hours power evaluated from 0 to 4. And when we apply the fundamental theorem of calculus and evaluate, we find that our surface area is going to be equal to pry or six times 17 Route 17 minus one square units no

Taylor Shimono · Answer

So for this problem, we want to find the surface area of the volume generated by rotating the equation where ice equal to the square root of four minus expert about the X axis from X equals 02 X equals two. So I know that we&#x27;re going to be able to apply our surface area formula, which tells us that her service area is equal to the integral from A to B of two part times FX times the square root of one plus and throw of X squared DX I&#x27;m so we&#x27;re given and B I wore a zero and B is two in our equation. In the last piece of information we need to find his f prime of X squared s. So we know that f of X is equal to the square root of four minus X squared. So, using the chain roll, the derivative of our radical is going to give us one over two times that square root. And then we&#x27;re going to multiply by the derivative of the inside portion, which is just negative to X. I&#x27;m so these negative twos are going to cancel and we&#x27;re just left with Ephraim Vex is equal to negative eggs over the square root of four minus X squared. And lastly, we want to find f problem X squared to plug into our collision, which is just going to be X squared over four minus X squared. And now we have all of our missing pieces of information for our integral. So our surface area is going to be equal to the integral firms. You&#x27;re in it too, of two plaque times F max with the square root of four minus X squared times the square root of one plus effort of X squared, which we found to be X squared over four minus X squared. Yes, and right now our interests looks fairly complicated, but it&#x27;s actually going to be pretty easy to simplify if we keep our square root of four minus X squared the same and our second radical on, we&#x27;re going to combine into one fraction, and it&#x27;s just going to be four over four minus X squared so we can see that these two radicals are going to cancel and scored four interest to So our surface area is equal to the interval from 0 to 2 for pie and then there should be a DX altogether. So when we integrate, this is going to be four pi X evaluated from zero to And when we apply the fundamental theorem of calculus, we&#x27;re going to find that our surface area is equal to eight pi square units.

Problem

Sketch the indicated curves by the methods of thi…

Problem 53 Easy Difficulty

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Caleb E.

Kristen K.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

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Basic Technical Mathematics with Calculus

Grace H.

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Grace H.

Caleb E.

Kristen K.

Michael J.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus