in-the-following-exercises-solve-the-given-maximum-and-minimum-problems-a-rectangular-hole-is-to-be-

Question

In the following exercises, solve the given maximum and minimum problems.
A rectangular hole is to be cut in a wall for a vent. If the perimeter of the hole is 48 in. and the length of the diagonal is a minimum, what are the dimensions of the hole?

Sherrie Fenner · Accepted Answer

question 13 Uh, here were given. That area equals 49 square. When we have area, we know that our to Gary Bulls are both positive. So we know that lane finds or it&#x27;s weird way also are know that our perimeter is our maximum and the formula for perimeter of a rectangle is to l plus two w I&#x27;m gonna solve my perimeter, sir. I&#x27;m going to solve my area. My length, times with himself for l. I have l u 49 over w. And I&#x27;m just ignoring the feet square right now. Um, and I know my answer will be in feet. So now I&#x27;m gonna take this equation here more racy and l sub secure in my perimeter. I&#x27;m gonna have a perimeter two times. Wants to get you. So this gives me a perimeter. Walls 98 we&#x27;ll double you plus double. And now, um, I want to, but I know that perimeter is my maximum so that when I take the derivative, I&#x27;m gonna get a zero. But first I like to write the w. I&#x27;d like to write the denominator as a negative exponents. I&#x27;m just gonna write groom ITER equals 98 of new negative plus two. And now when I take my derivative, I am get Oh zero equals because my maximum negative 98 that&#x27;s I have a 98 over W Square plus two. And when I saw or my w attractive moods I thio weird thank you to use squared two and you were 49 thing 49 plus reminding us in this case we&#x27;re only looking at areas has value. So looking at seven and my units are in the my with his seven, my length would also have to eat seven because seven times that, you know

Matthew Lee · Answer

for this rectangle. Uh, he was directing like this. When I say we don&#x27;t know the dimensions what we call the fight. I&#x27;m next in line. You know that the area should be equal to a That means that Aye, aye, Go, ext. Time to wine and we&#x27;re trying to find a minimum perimeter. Oh, the question for perimeter put you I heard about single on. If this is what we like to minimize, and we need to write it at the function of a single variable. What do you want to write to the function of acts? You can do that by you? Can other equation of every up here helping for why? And then walking back into this equation that only left with only accepted his offer? Why? Why go over X number eight? A constant. And if we plug down bedding down here, we&#x27;ll get to Rex two. Damn right you&#x27;re Yeah. Hey, over it All Right. Now, we need to find the minimum of this function here. But to do that, we need to find the critical number, which means we need to first derivative. Let&#x27;s take the derivative of the function. You can have power over, You know, to Huh? Uh, this is excellent. Negative first power. So it&#x27;s going to get the negative time Negative, too, right? The next squared. All right. The minimum will be where the critical numbers are. Uh, so critical number there when that you&#x27;re riveted equals zero very court of your own narrative, too. What? It was a financial minus. Now you have in mind if I&#x27;m the fraction so my ah, to a over X squared. Okay, copper axe. Then add this expression over to the other side. You get to a I&#x27;m over x squared cause too. Um, yeah, I think we&#x27;re back to the right side by modifying my actuary Jack&#x27;s quick cherie, uh, actual ego. Bye bye. Chewing and take incorporate puts in my for today. Now, however, some physical quantity it is the weight of this rectangle ex can&#x27;t be negative. That means actually closed for today. It&#x27;s the only critical number and the only critical number. It must be the number or we have the minimum perimeter. Our function the only possible number. Uh, so we just found explore the dimensions of this rectangle. Find the other dimension to find out why I just knew this equation we had earlier Michael gave Rex. Uh oh. Not for today. So why would a divided by grew today? And when you take a number and divide by square root where dismember this quarter, so exit with growth of a nationwide with grow today the rectangle which has what? The fix area that has a minimum perimeter and those two numbers of the same way that we actually get a square in the case. So when we could think of a square is that he thinks the area dick fury inside here and directing, though that has a minimum perimeter. You don&#x27;t mind a perimeter to cover this thing?

Jessica Hill · Answer

in this problem were asked to solve for the length and width of this rectangle. And we have a perimeter that we know. It&#x27;s 48 centimeters, and we know that the wets this three centimeters shorter than half of the lakes. So let&#x27;s go ahead and start everything sort of rectangle down in the lower left hand corner. So we know that the width is three centimeters shorter than 1/2 of the length. So I&#x27;m going to use this side as my woods. Well, both of these sides in bluish my with, um so we know that it&#x27;s three centimeters shorter than half of the length. So that means if I take half of the length and subtract three centimeters from it, that will give me my width. And I can go ahead and just represent my links using l. So you fill in all four sides if I would like and I know that two of my legs and two of my woods added together are going to equal my perimeter of 48 centimeters. So I&#x27;m going to write that out into an equation. So two of my links, or two times l plus two times my with 1/2 l minus three equals my perimeter, which I have been given as 48. So let&#x27;s go ahead in solve our equation. So first, I&#x27;m going to distribute my to into my princes here, so I&#x27;m going to simplify this 20 plus two times. 1/2 l is just going to give me one. No, or l two times negative three is going to leave me with a negative six equals 48. Okay, so now I can combine like terms on my left side here of two l variables. So two al plus one l use me three. L&#x27;s minus six equals 48. Now I&#x27;m going to take my six and Abit onto both sides. No, Leave me with three l on my left and 48 plus six when you leave me with 54 Now I&#x27;m going to do my friend of step to figure out my length. That is going Teoh be to divide by three Thais late my variable. So 54 by the by three Whose me 18. So I know that my length is equal to 18. So that&#x27;s part of my answer. But let&#x27;s figure out my with. So I know that my width is 1/2. I&#x27;m going to do it in green here. What is 1/2 times my length, which now we know is 18 my ass. Three centimeters. So half of 18 is nine. Then all we have left to do is nine minus three, which is going to leave me with six. And my width is equal to six. Okay, so now we&#x27;re just gonna go ahead and add in our labels, and they&#x27;re in centimeters. So six centimeters from my with 18 centimeters for my length, and that is my solution.

Amrita Bhasin · Answer

okay, We know that the the volume is linked terms with times height, so volume equals, we know the length is four months X minus X, which is four minutes to x, and we know the with is to my sex minus X, which is to minus two acts. So we&#x27;ve got four months to ex temps, two miles to X and then Times X because long times with times height, which high is just facts. Okay, this is equivalent to have you distribute for X cubed minus 12 X squared, plus eight acts. Now, given this, we know from the first derivative tax test that we end up with factoring four times three x squared, minus six x plus two said equal to zero. You explain. My six x plus zero plus two equals zero. You can use a quadratic formula for this, and you would end up with X equals one minus square of three over three. Okay, so now that we know, this means that the height height X equals one minus squared of three over three Put me into the length Formula four minus two x, we end up with length is two plus two square root of three over three and now looking at the with the formula for this was to minus two X. We end up with to minus two plus two squared of three over twos. That&#x27;s just juice. Word of three over two over three, two squared of three over three.

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

Problem

In the following exercises, solve the given maxim…

Problem 15 Easy Difficulty

In the following exercises, solve the given maximum and minimum problems.
A rectangular hole is to be cut in a wall for a vent. If the perimeter of the hole is 48 in. and the length of the diagonal is a minimum, what are the dimensions of the hole?

Answer

Related Courses

Basic Technical Mathematics with Calculus

Related Topics

Discussion

Top Calculus 1 / AB Educators

Heather Z.

Caleb E.

Kristen K.

Samuel H.

Calculus 1 / AB Courses

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 24

Video Transcript

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Heather Z.

Caleb E.

Kristen K.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

Basic Technical Mathematics with Calculus

Heather Z.

Caleb E.

Kristen K.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

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

Heather Z.

Caleb E.

Kristen K.

Samuel H.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus