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(a) A wholesale manufacturer of canned corn wants a tin can (a right circular cylinder) that will have a volume of 54 cubic inches. As little tin as possible as possible is to be used in the construction of the can. Determine its height and diameter. (b) Measure a can of com from your local supermarket. Does its dimensions yield a minimum? (c) Why do you think manufacturers choose not to use this type of design?

(a) $h=2 r ; r=\frac{3}{\sqrt[3]{\pi}}$(c) doesn't look sleek.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

Missouri State University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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find the can of minimum surface area with volume 54 cubic inches. How does that compare to a real can of corn or can of anything? Alright, so first, let's look at what a can would look like if you cut it apart. It would have a top. It would have a side which would be in the shape of a rectangle. So a lateral surface, and then it would have a bottom. Okay, I'm gonna fix that top there. Okay, so we're trying to make make this out of the minimum amount of tin or whatever it's made out of. Okay, So what we need on the top and bottom is the radius. And then what we need on the rectangle are the length and the width. Okay, let's call this H for height, height of the can. And then this distance right here has to go all the way around the circle. So it's the circumference of the circle. Two pi r to So we're trying to minimize the surface area, which he wants a top by r squared ah, bottom higher squared and the lateral surface or the side two pi r times H or two pi r h so two pi r squared plus two pi R h well, to minimize anything, you take the derivative, you set it equal to zero you solve. The problem is, we have too many variables here, so we need a constraint on our constraint. Is this the volume is 54 So constraint, What? Wow, I don't know what That That's just weird that strike it constraint. Okay, the volume of ah, cylinder or a can is area of the base fire squared times the height. So think of it as piling up potato chips or something. Or Oreo cookies. Okay, One of them, two of them. Three of how you wanna make them. So the area of the circle fire squared times Age has to be 54. Okay, so now we're going to solve this Pires court h equals 50 for for either Rh plug it into the Southern equation and then take the derivative were clearly solving for H will be easy. Plus, it'll be easy to plug in, so h will be 54 over pi r squared. Okay, so surface area equals two pi r squared plus two pi r times age which is 54 over pi r squared. So two pi r squared. Plus you're the pies cancel and one of the ours canceled. And 2054 that's 108 over our or times are to the minus one. And I wrote it that way so that I could take the derivative faster. Here it goes. Um, two pi. That's a constant. So two pi times the derivative of our square, which is you are plus 108 times the derivative of art to the negative one, which was negative are to the negative too. So four pi r minus 108 over r squared equals zero. So I changed it to a negative. Exponents take its derivative, But then, to solve our simplify, I must always use fractions. Okay, so I'm gonna move one of them over for pi r equals 108 over r squared, cross multiply. They are over here and the four pi over there. So our cube equals 108 over four pi for goes into 108 27 times S o r. Take the cube root of both sides. Three over the cube root of pi and and then h is remember 54 over pi r squared So 54 over pi times three over Cooperative pi squared. Okay, which looks pretty nasty, but it's not that bad. Case to get 54 pi times nine over pie to the two thirds power. So that's six over pie to the one third power or six over the square root of pi. So that's our is three over the cube root of pi and H is six over the cube root of pi upset, So h is twice harm. Okay, so I went and I looked at a real can and then I drew a picture so real can Looks like this. Um, it's three times the radius. Okay, the height. This If this is our this site is three are And then I drew another can of minimal surface area, but with the same volume of this one. And this one is our and this one is two. Different are, though, because our to our Okay, So why do you think they don't use this can of minimal surface area if it's going to save the money? Well, in the book, it just says, because it's not as attractive or not asleep looking. But I don't really know. I guess that's an opinion question.

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