### Answer

Optimal soda cana. Classical problem Find the radius and height of a cylindrical soda can with a volume of $354 mathrm{cm}^{3}$ that minimize the surface area.b. Real problem Compare your answer in part (a) to a real soda can, which has a volume of $354 mathrm{cm}^{3},$ a radius of $3.1 mathrm{cm},$ and a height of $12.0 mathrm{cm},$ to conclude that real soda cans do not seem to have an optimal design. Then use the fact that real soda cans have a double thickness in their top and bottom surfaces to find the radius and height that minimizes the surface area of a real can (the surface areas of the top and bottom are now twice their values in part (a)). Are these dimensions closer to the dimensions of a real soda can?

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### Video Transcript

okay for part A were given a constraint that the volume of a cylinder is equal to 354. Okay. And the volume of a cylinder is equal to pi r squared times height. Now we want to minimize ah surface area so the surface area is gonna be two times pi r squared, Uh, plus two pi r for the circumference times high to give us a service area like the sheath or whatever. Okay, So if we solve for H and R volume equation, uh, we're gonna have ages equal to 3 54 divided by pi R square So you can plug in for H and R equation here because our new service area equation will be, uh, two pi r Square plus 6 90 The pies will cancel and we're just left with one are in the denominator. Okay. Now, to find, uh, Teoh, um, minimize our surface area. We're just gonna set, um, derivative equals zero. We're gonna have four pi r tu minus 6 90 over our square. I add 6 92 but over our square to both sides and multiply by r squared. I'll have 6 90 is equal to four pi r cubed on our it would be the cubrir uh, well, 6 90 divided about four pine. I just got through that and calculator real quick. We'll have a radius of around 3.8. Okay, If we plug that back into our height, that's gonna give us a height. Mrs. Put that in. So three before about it by it comes our square.

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But is that a 3.8? Now? There are high. It's gonna be around 7.8. Okay. And so Part B tells us to look at a real soda can, Uh, it gives us that the real set of can the radius is closer to 3.1, and the height is closer to 12 only they have new parameters where they're gonna have double thickness in the top and bottom. So same thing for the volume. We already have our age soft for we already. Um, but our surface area, instead of multiplying, is that them? Two pi r times h. We're gonna have four pi r times age. Um, that's really the only difference we're gonna have. Sorry, Derivative. Too much all comes out the same. We're gonna have two Pi R Square Oh, wait, That's our that four pi r And that's gonna be for the I don't think I've overthought myself. Ah, the part of the surface area that has the high is the sheets. So I'm not worried about that. We're gonna We're gonna do two pi r square to make it four pi r squared and then bringing down the two is gonna make that eight pi r that accounts for the top and bottom and then we're still gonna have plus 6 90 about about our king up 6 90 minus paper I minus 6 90 over r squared. Okay, so again, following the same process will have 6 90 is a is equal to eight pi r cubed and therefore are is equal to the Q brew 06 90 divided by a pie. Okay, can going to plug this into the calculator. We're gonna have 3.2 for a radius and then for our, uh, all right, have clove 0.38 which is a lot closer to what the actual, um, dimensions are. And these dimensions are closer to the dimensions of a riel can. Good. Thanks very much.

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