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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?

Calculus 1 / AB

Chapter 24

Applications of the Derivative

Section 7

Applied Maximum and Minimum Problems

Derivatives

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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'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'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'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're gonna say time's too. Why so volume eyes able to So let'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'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're setting that equal to zero. So we have, um, six pie. Why? Squared is going to be equal to two pi. We'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'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're done.

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The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

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