in-the-following-exercises-solve-the-given-maximum-and-minimum-problems-referring-to-exercise-24-wha

Question

In the following exercises, solve the given maximum and minimum problems.
Referring to Exercise $24,$ what are the radius, length, and volume of the largest cylindrical package that may be sent through the mail?

Mike Gaerlan · Accepted Answer

Okay, so here we are, given a sphere. Um, And inside that sphere were given a, um we need to fit a right circular cylinder inside this sphere here. So that looks like this here. Roughly speaking. Um, so let&#x27;s just take a cross section of that sphere, right? Like so. Like, here. Okay. So we know that the radius of this fear is he with you one? Um, so we need to find the the radius of the right cylinder and as well as its hype. Um, the height is going to be, uh, h but I&#x27;m gonna call this height. Why here? So we can rewrite Why, As a function of our as well, So we have. Ah, One squared is equal to r squared plus y squared since in sin side of this sphere. Okay, so we can rewrite r squared is going to be equal to one minus y squared. Okay. And then also H is equal to to Why. So remember the volume of a all right circular cylinder is pi r squared times. H. Right. Uh, so in our case, it&#x27;s going to be pie. And remember, So we just found that are square it is able to one minus y squared, right? So we just plug down in here and then h is too. Why? So we&#x27;re gonna say time&#x27;s too. Why so volume eyes able to So let&#x27;s simplify that a little bit too. Why? To py y times one minus y squared. And we can expand that out to pie. Why minus two pie? Why, Cube? So to optimize this, we&#x27;re going me to find Devi, um, and set that equal to zero. All right, So Devi d y here is going to be equal to two pi minus six pie times y squared. Okay. And so we&#x27;re setting that equal to zero. So we have, um, six pie. Why? Squared is going to be equal to two pi. We&#x27;ll divide by two pi on both sides, and we get three y squared is equal to one. So why is going to be equal to the square root of one over three here? Okay, so that is why so now to find volume in terms of why, right? That is going to be, uh, so remember, volume is equal to two pi. Why? Times one minus y squared, right? So why is equal to square root of one over three. Right? So volume is going to be equal to to pi times the square root of one over three times one minus the square root of the square root of one over three squared. Right. Um, so that&#x27;s going to be two pi divided by just divide by square 23 times one minus 1/3. Right. Since the squared of one over 30 squared. Um, is this gonna be one over three? Um, so this becomes two pi over a square root of three times to over three. So our total answer is gonna be four pi over three square root of three, and then we&#x27;re done.

Sherrie Fenner · Answer

every question. 29. We have a rectangular box and our and is a square. So our volume is gonna be X squared times why I cannot draw three dimensional. Um, and we know that our is 100. Just the perimeter is equal. Uh, why plus two x plus two x so 108 inches. Or we could say why he won&#x27;t 108 lines for X. And now I&#x27;m gonna put this 108 minutes for X back into my volume. So we is going to eat more. Thanks were times a minus X which gives me on and the Eaton&#x27;s. I want a X weird minus four x cute. And I want to take the derivative of that to find the organization for Have the room, the Coons. 216. Thanks, minus 12 x there. And now we just solve it like we would go and, um, Rambla or any quadratic my I guess the easiest. I think I&#x27;m just gonna write it over here isn&#x27;t gonna factor out in X shame on the factor. Yeah, just thanks. So either X equals zero or to 16 minus 12. X equals zero. So negative. 12 checks Negative too. 16 by both. I&#x27;ve been negative. 12 therefore expertise for 18 inches. When you plug that back into your y equals you get why equals one? 08 minus four times 18. Yeah, for why peoples?

Mike Gaerlan · Answer

Okay, so we have a cylinder. Who&#x27;s volume is going to be be, uh, his volume is going to be 16 pie. So remember that the formula for the volume of a cylinder is also equal to a circular. Cylinders people do pi r squared. H, right. Um, we need to minimize the amount of surface area. Surface area has the formula. Right? So remember the ah, the bottom of this cylinder has formula pi r squared. So we need to double that since there are the top in the bottom. So we need two pi r squared and then plus so the the other face this one here, right? It has a formula two pi r times h two pi r. So that&#x27;s the, um two pi r is the circumference of the of the circle at the bottom and then times the height Okay, um, so let&#x27;s solve for H here using this. So if he saw for age, we get that, um, eso 16 pie is equal to pi r squared h So you divide by pi r squared. We get that h is going to be equal to 16 divided by our square here. So let&#x27;s plug that into our surface area. So our surface area is going to be two pi r squared plus two pi r times 16 over our square, which can simplify to two pi r squared plus, uh, 32 pie divided by our okay, which can also be written as two pi r squared plus 32 pi r to the negative one. Okay, Now we need to minimize the surface area. So we&#x27;re gonna take the derivative of the surface area with respect to our So the of essay delivered to the service area. With respect to our this is going to be equal to four pi r minus 32 pi times are to the negative too, which is also equal to four pi r minus 32 high divided by r squared To minimize it, we need to, um, set that equal to zero and then saw for our So we have four pi four pi r four pi r is equal to 32 pie fight by r squared If we multiply by our squared on both sides. Okay, move that over. So flying four pi r is equal to 32 pi over r squared That&#x27;s right. R squared, multiplied by pi r squared on both sides We get for pi R cubed is equal to 32 pi. Okay, divide by four pi on both sides. We get that our cube is going to be equal to eight. Um, taking the cube root of eight. Get that are must be equal to two. Okay. And remember what h must be equal to ages is H is equal to 16 over r squared. Right? So if you take that, um, H is equal to 16 over r squared, so H is equal to 16 over two squared, which is 16 over four. So we have that h is equal to four. So our dimensions of our circular cylinder is that our equals two and h equals four. Um, and that is our final answer where you&#x27;re done.

Amrita Bhasin · Answer

Okay, We know that the triangle&#x27;s a B C and a D e r. Similar. Therefore, given what we know, we have h minus capital H minus lowercase h over our equals capital H over our capital R. Now this means that writing this in terms of lower case H, we have lowercase H equals capital. H times one minus are divided by capital R. Given this, we now know the volume. The cylinder is pi r squared h so writing the volume in terms of lower case and capital H, we have pi times Capital age divide by capital R times are times lower case R squared minus r cubed Okay, Taking the first derivative and setting it equal to zero we end up with are times two are minus three are equal zero giving us our equals zero and our equals 2/3 Thames Capital R. Okay, now, given this, we have to lastly take the second derivative. So we&#x27;ve got pie age Divide by capital are times two are minus six are giving us negative two pi h divide by capital. Our times are this is negative. We know it&#x27;s less than zero. This implies from this that the maximum volume is gonna be We&#x27;re looking at the formula pi r squared h so literally plug in what we&#x27;ve figured out and this simplifies to four pi divide by 27 h times are squared equals v

Amrita Bhasin · Answer

okay, We know that we have a right triangle. ABC is a right triangle. Now, given this, we know that we have the equation. H q equals R squared or radius squared in the context of the stamina of the sphere. Because we&#x27;re looking at the sphere. We know that diameters double the radius, so we know h plus Q equals too. So we can write this as a TSH times two minus h equals R square. Remember, we want to eliminate as many variables as possible. The volume of cone is 1/3 pyre. Squares weaken right. This as one third pie h times two minus h times h Because remember we wrote I score. We were are scored in terms of h, so we&#x27;re literally just plugging it down here. As you can see, 1/3 pi r squared h Okay, now, obviously, this simplifies to 2/3 h squared minus 2/3 h cubed. Okay, Now let&#x27;s take the first derivative V Prime of X equals zero. So starting this equal to zero, we have 2/3 minus h times each said equal to zero giving us H equals zero comma. 2/3. Okay. Now H equals zero. The volume is vehicle zero, the minimum value. So taking the second derivative, we end up with 2/3 minus four h. Plugging in H equals 2/3. We end up with 2/3 minus 8/3. So given this, we know that the volumes at a maximum when h equals 2/3. So plug in a final time pi over three times 2/3 squared times to minus 2/3 and we end up with four. Pi over 81 equals our volume.

Problem

In the following exercises, solve the given maxim…

Problem 25 Easy Difficulty

In the following exercises, solve the given maximum and minimum problems.
Referring to Exercise $24,$ what are the radius, length, and volume of the largest cylindrical package that may be sent through the mail?

Answer

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Precalculus Review - Intro

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

Grace H.

Anna Marie V.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus