in-the-following-exercises-solve-the-given-maximum-and-minimum-problems-a-window-is-designed-in-the-

Question

In the following exercises, solve the given maximum and minimum problems.
A window is designed in the shape of a rectangle surmounted by a trapezoid, each of whose legs and upper base are one-half of the base of the rectangle. See Fig. $24.64 .$ If the perimeter of the window is $5.70 \mathrm{m}$, find the width $w$ and height $h$ such that it lets in the maximum amount
of light.

Linda Hand · Accepted Answer

Yeah, okay, here&#x27;s a norman window, it&#x27;s a rectangle with a half circle on the top and we know that the perimeter is 30 ft and we want to find the dimensions of the window that lead in the most light. So that means we want to maximize the area. So we need a formula for area of this. It will be easy with the area of a rectangle plus area about half a half a circle. But we&#x27;ve got to name it cleverly. So let&#x27;s Let&#x27;s call the radius here to be x radius of the semicircle. So that means this side is two x. And then let&#x27;s call these sides. Why? Since the vertical? And then if the radius is X, we need the perimeter of the circum half the circumference of a circle. So uh circumference is two pi R of a circle. So in this case circumference is two pi X but we only have half a circle, so one PX Okay, so then the perimeter is going to be this. Why? Plus pi X. That&#x27;s why plus pi X plus Y plus two X. Okay, so that&#x27;s our constraint constraint, Y plus pi X plus Y plus two X. Has to Equal 30. Okay, or two Y plus two plus pi X equals 30. And then we&#x27;re trying to maximize area. So a equals well it&#x27;s a rectangle. So it&#x27;s areas linked times with so two X. Y plus the area of half a circle. So remember the area of the circle is pi r squared. So in our case pi X squared and we need half of that. So half the area is half pi X square. Okay, So to find the maximum of anything, you take the derivative and you set it equal to zero and then you solve and then you answer the question Okay, But we have a problem here in our problem is we have too many variables. We have X, Y and X squared. So we need to get one of them out of there and that&#x27;s what the constraint is for. Okay, so now you decide do you want to solve it for Y or do you want to solve it for X? And then after you do that, which one do you want to plug in? Well, I&#x27;m solving for Y. I get to Y equals 30 minus two plus pi X. and so why is 15 -2 Plus pipe X oops. Two plus pi x oops. Mhm. Two plus pi X over two. Mhm. So now my area formula is two times X Times Y. Which is 15 -2 plus pi X over two plus one half by X square. So if I multiply two extra here, I get 30 X plus two plus pi X squared plus one half pi X squared. So I have 30 X plus two X squared plus pi X squared plus one half pi X squared. So one plus a half. That&#x27;s three halves. So now I have 30 x Plus two x squared plus three halves, pi X squared. All right, That&#x27;s a so now I&#x27;m gonna take the derivative with respect to X. So it&#x27;s 30 plus for X. I&#x27;ve made a mistake. Right here, That&#x27;s 30 x two plus pi X squared. And so that is 30 X minus two X squared -909ared. So that is 30 X minus two X squared -1 X half pi X squared. Excuse me? The way I knew that I made a mistake is because I was going to get all positives here and I wasn&#x27;t gonna be able to set it equal to zero. Alright. Here we go again. Sorry about that. All right. Going to take the derivative derivative of 30 x. 30 minus for X -1/2 pie times two X. So 30 -4 x -9 X equals zero. So 30 equals 4 -9 times X. 04 plus pi So x needs to be 30 over. Okay, sorry four plus pi And then remember we had why was Why was 15 -2 plus pi X over two. So why is 15 two plus pi Over two times 30/4 plus pi. Okay that two cancels with that and give you 15. So now it&#x27;s 15 times 1 -2 plus pi Over 4-plus pie. I don&#x27;t know why I&#x27;m going crooked there. Okay, so um that&#x27;s 15. Going to get a common denominator four plus pi That&#x27;s four plus pi minus parentheses. T two plus pi Over four Plus pipe. So 15 times to over four plus by or 30/4 plus one. Hi so the dimensions are this Okay. X. is 30/4 plus pi. So make the base 60/4 plus by and then make the verticals. Whatever. Why was I forgot already? Um 30/4 plus. Why four Plus by Army. Okay. If you don&#x27;t like that, change into decimals. Okay so the trick was first you have to get a good constraint and you had before that you had give everything good names. I could have called the base X and then the radius would have been one half X. And then I would have a bunch more fractions and clearly that would not have been good. Okay, so name everything correctly, write your constraint. Figure out what you&#x27;re going to maximize takes to take the derivative saturday equal to zero. Super fun time.

Mike Gaerlan · Answer

Okay, so we have a triangle with the sides. Um, X equals zero. Why equals zero and X, divided by four. Plus why divided by six is equal to one. All right, this has, um, a, uh, intercept here at the 0.40 Right, cause if y equals zero all right. Y equals zero X must then equal to four. Right? And then this also has an intercept here at six. Sorry. Zero zero comma six zero comma. Six. Where if What are Sorry if X equals zero, then why must equal six? Right? So we&#x27;re trying to optimize the area of a rectangle that fits inside. Uh, this this triangle. Right? So the formula for this rectangle is essentially going to be a is equal to x times. Why? Where x and y are going to fall on the line. So we&#x27;re gonna have X divided by four. Plus why divided by six is equal to one, um, multiplying by 24 on both sides. For the for the constraint, we get something a little simpler. Six x plus four. Why is equal to 24 So solving for why we get that why is able to 1/4 times 24 minus six x Talking this into our formula for the area we got. That area is equal to X times. Ah, All right. Time&#x27;s 1/4 24 minus six x or in total area. You see the two 1/4 for simplifying six X squared. Like so. So now we need to find the derivative. So rewriting the area is equal to 1/4. 1/4 24 X minus six X squared. Right. So we need to find the derivative of the area with respect to X. This is going to be 1/4 times 24 minus 12 x. Remember, we&#x27;re sending that equals zero. So we get that 24 minus 12 x is equal to zero or 12 X is equal to 24. So ex is equal to two now, Um, why here then? Um, we plug in. If you look back here, uh, why is equal to 1/4? 1/4 24 minus six x right, 24 minus six x. So why is going to be equal to 24 minus six times two. So 1/4 of, uh, 12. So this is going to be three in the area. It&#x27;s gonna be two times three. So our area is going to be six. So this is our answer. Uh, here six.

Darshan Maheshwari · Answer

problem. Number 52. The acumen dark. There&#x27;s a building from its first to be known, the building with a straight line services hold. The building is Let&#x27;s suppose this is the ground. Okay, so 8 ft long fencing runs battler to the building. So let&#x27;s say the fence singers, they combined it. So this is a we don&#x27;t know the height of the building that the height of the buildings in open by edge. So over here we have Burling. This is friends on that This just between the building and offenses four. So this is for okay. So we have to find the minimum length of the ladder, which will drum just above this fans fence and official test a building. So that fear talking about the minimum ladder the next just touch the fence. And really, it&#x27;s the really So that&#x27;s how the fence will look like in the green. You have to find them a new homeland defense. So let&#x27;s let this land be X. Who here? So if you look at these two triangles is gonna be approved Wranglers devoted by a that this would be that BC it has been deemed that be so these two triangles are actually similar because angle is common on. Then we have angle be an Anglesey is named illegally using the properties of similarity of the triangle. This we have. We can write that you can write a relation which is he See over a scene will be good baby or beer. This is because he&#x27;ll be d trying similar trying the a c t by e similarity This meat are easiest edge. He&#x27;s he is express for the Murdoch more DB, which is eight and behave, which is X. So from here begin W H has eight times experts four words is the leverage on what we have to minimize. We have to minimise the limp off the ladder, which is a so if you look at triangle a ce, is he trying is actually forming a right angle trying where were interested in each of the length of the ladder using by the worst year we have these where will be a C square unless he sees work. So from here, the value of gays for these e square comes orders A C is express for So we have experts for square over here and then you talk about easy, which is edge So the red square. So we have a a square as explosive for whole SquarePants. We have actual here. Values this. So let&#x27;s substitute dark. So we have eight x plus for over Rex, we&#x27;ll square from here. We have explosives for rules were plus 64 over X square experts four whole square. Then we taken in the route as well. This is the length of the ladder 80 which we have to minimize using a graphic It graphically, calculatedly plodders. And its graph looks like this. Clearly, we can see that when the value off the variable X is 6.35 than the length of the ladders minimum on the length off the ladder, which is minimum. He has given us to 77.1 for it On the unit of the ladder would be feet, because there other units are also gonna speak. So this is the minimum rental. A lot of

Karly Williams · Answer

hi, everyone. So today we need to find the maximum and minimum areas of each rectangle. These air the measurements for a rectangle, so it&#x27;s measured to the nearest unit. This tells us that are possible. Error is half of a meter. So for our minimum, we&#x27;re going to subtract 1/2 from each of these. So we have 6.5 meters by 7.5 meters. Then for our max, we are going to add the half. So we have 7.5 meters and then 8.5 meters. Now we&#x27;re going to multiply all of these. So we have seven times eight gives us 56 6.5 times 7.5 gives us 48.75 meters or 48 in 3/4 meters. Then we have 7.5 times 8.5, which gives us 63 0.75 meters. So these are the areas for all of our measurements. So 48 in 3/4 meters is our minimum, and 63 3/4 meters is our Maxwell

Karly Williams · Answer

Hi, everyone. So today we need to find the minimum and maximum possible areas for each measure. These are a rounded to the nearest whole measure, so that means we&#x27;re going to have half a foot of difference for the minimum and maximum. So for a minimum, we&#x27;re going to subtract half a foot, so we&#x27;re gonna have 23.5 feet in 21.5 feet. Then for the maximum, we&#x27;re going to add it. So we&#x27;re gonna 24 late fall in 22.5. All right. Won&#x27;t. Now, to get the areas, we have to multiply these together. So we get 528 feet squared for a regular area. For our minimum, we get 505 points, 25 feet squared. And for our maximum, we get 551 point 25 feet squared for our maximum

Linda Hand · Answer

Okay, here we have a Norman window, and that means a rectangle with a semi circle on the top. And we know that the perimeter of this window, including this round part, is 30 ft. And they want us to find the dimensions of the largest window or the one that look, Let&#x27;s in the most light, which would be the largest. Okay, largest. So we&#x27;re trying to maximize what? We&#x27;re trying to make the largest windows. So we&#x27;re trying to maximize the area of the window and the area would be area of the rectangle. Okay, plus the area of the semicircle. Just say half circle here. All right, so let&#x27;s get some labels here. Let&#x27;s call the height of the window. Why? And then let&#x27;s call the Okay, So here&#x27;s what I&#x27;m thinking about. Do I want the Radius to be X or do I want the length here to be ex? I&#x27;m gonna make the Radius X that white don&#x27;t have to mess with the fractions there. So then this is two X. Yeah, All right, so now we can do it. The area of the rectangle is two x times y plus the area of a half circle its half pi r squared in rs X here. All right, so we&#x27;re gonna take the derivative of that instead of equal a zero and solve. Okay, But it has too many variables in its We gotta figure out how x and y are related. And that&#x27;s why they gave us the perimeter. That&#x27;s our constraint right there. Constraint. P equals 30. All right, so we have for the perimeter, we have Why, plus two x plus why? Plus not to x. We don&#x27;t need this across here, Okay? Because that&#x27;s not part of the perimeter. What we need is this right here, which is half the circumference of a circle have circumference. That&#x27;s half of two pi r. So that&#x27;s pi r and R in this case X Hi, ex. So here we go again. Why? Plus two x plus why plus pi ACC&#x27;s and that has to equal 30. Okay, so two y plus two x plus pi X equals 30 trying to solve for y because that will be the easiest thing to plug in. What? So to y equals 30 minus two x minus pi X multiplied by two. Okay, so then the area Okay, two x times. Okay, that was that. Two X Now we&#x27;re putting in. Why? Which is this? 30 minus two x minus pi X over to plus one half pie export. Now, I&#x27;m really glad I made the Radius X because now this too, And this to cancels. So I&#x27;m gonna go ahead and multiply out here. So I have 30 X minus two X squared minus pi X squared plus one half pi X squared. So that&#x27;s 30 x minus two X squared minus pi X square plus a half pi x squared that b minus a half pi expert That&#x27;s the area in terms of X. Now I&#x27;m going to take the derivative 30 minus four x minus one half pi times the derivative of X squared, which is two x so 30 minus four x minus I X. I want that Teoh equals zero. So 30 equals four x plus pi acc&#x27;s factor Net X out here for plus by so x is 30/4 plus pi Can one of my trying to find find the dimensions of the largest window? So now I need to know what? Why is why is 30 minus? I forgot already. Two x minus pi X for two You? Yeah. So 30 minus to plus pi over our times 30/4 plus pi all over to So that&#x27;s 30 times four plus pi minus to plus pi times 30 all over four plus pi Oh, To do that, I&#x27;m multiplied everything by four plus pi in the to clear off the fractions Okay, so 1 20 plus 30 pie minus 60 minus 30 pie over four plus pi So 60/4 plus pi So x waas 30/4 plus pi Oh, convert that to a decimal sort. Sounds like a number two you four plus high divide by 30 Reciprocal So X is 4.2 And why is twice that much? 8.4 because it&#x27;s 60/4 plus pi. Okay, so the dimensions are, um why which is 8.4 and the base here 8.4? Because there will be two times two times what x waas so may make it 8.4 tall on the rectangle part 8.4 wide on the rectangle part and a semi circle of radius 4.2. So that&#x27;s how you going to do it so that the perimeter is 30

Linda Hand · Answer

We know that if X is half the West and why is the height, then we know the perimeter would be, too. Why plus two plus pie times acts. And we know this could be said equal to 30 because remember the perimeters 30. Therefore, we know we can write why there&#x27;s 15 minus parentheses, one plus pi over too. Times acts okay. The area of the window is the area of a rectangle, plus the area of the semi circle or half the circle, which is to pie. Why? Because pi X squared divide. But you we know that we have a of acts. There&#x27;s 30 acts minus two. Close pi over too terms X squared and a prime of axe. So they end up with Axel into this equal to zero access approximately 4.2. Therefore, we know the area is a maximum went ax is 30 over four plus pock, and we know this is approximately 4.2 feet. So the radius of the semi circular part is 4.2 feet or 30 over for post pie. And then we know that the dimensions of the rectangular part are 8.4 times 4.2, so you just double it. So 8.4 feet times 4.2 feet

Linda Hand · Answer

The problem is a Norman window has the shape of a rectangle ce Monday some circle If the perimeter of the window is our defeat, blinded the dimensions of the window outside. The greatest possible amount of light is admitted friends to look at this picture if we light one side if they would act. Other side of the rectangle is here. Why behalf Primitive of the window is a call to ACS. Asked why plus hi, I&#x27;m over too. So this&#x27;s control thirty. Well, at a It&#x27;s the conscious area off this window. A If it hurt your X times why plus half one half times Hi grams axe over to score. This is area off half circle. This is in control. Thanks. Why plus hi. Thanks, squire. And what I need to do is to find the maximal while you&#x27;re off the first from this equation with half. Why is he called you? Well, half Pam&#x27;s thirty minus sex minus pi ax Over too. A week or two. Ex tam&#x27;s. Why, plus high ex squire overhit and take preventive. A promise is equal to well half terms. Thirty minus tax minus Hi Max over to bus ex terms. nicked One negative high over too class. Hi. Thanks. Over four Simplify it. Dysfunction we have This is Echo two Negative one plus Hi over four. Thanks plus fifteen. If it if we like a prom is equal to zero, we have X is equal to fifteen over long pass high over four. This is Echo two sixty cooler or us part. This&#x27;s apart. Eight on your fear. If axe is greeters and this number we can see a prom is last sons zero If axe is last times this number and from is good than zero So what axe is ico chou? Sixty are We&#x27;re full blast pie. Hey has a maximum value in this case. Why does he hurt you? One half terms thirty minus x. My spy homer packs over too. This is control. No, this&#x27;s about four point two zero. So we have one X is equal to eight point four theory. Why is a cultural point two zero The greatest possible months of light has admitted it

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In the following exercises, solve the given maxim…

Problem 40 Easy Difficulty

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Catherine R.

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Basic Technical Mathematics with Calculus