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anonymous
 3 years ago
A cylindrical can is made to hold 500 mL of soup. Determine the dimensions of the can that will minimize the amount of metal required.
anonymous
 3 years ago
A cylindrical can is made to hold 500 mL of soup. Determine the dimensions of the can that will minimize the amount of metal required.

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zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.2Volume of a Cylinder:\[\large V=\pi r^2h\] They want us to minimize the `amount of metal`. The amount of metal is the `Surface Area`. Surface Area of a Cylinder (If I'm remembering this correctly):\[\large A=2\pi r^2+2 \pi r h\]

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.2They want us to minimize the surface area, given a constraint on the volume.

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.2\[\large 500=\pi r^2h\]Solving for h gives us,\[\large \color{orangered}{h=\frac{500}{\pi r^2}}\] We'll plug this into our Area formula. \[\large A=2\pi r^2+2 \pi r \color{orangered}{h} \qquad\rightarrow\qquad A=2\pi r^2+2 \pi r \color{orangered}{\frac{500}{\pi r^2}}\]

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.2Simplify it down and then take the derivative of your area function. Then setting it equal to zero will allow you to find critical points, namely the value of r that will minimize the area.

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0\[\huge A=2 \pi r^2+1000\]

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0\[\huge 0=4\pi r1000r\]

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0am i doing it right so far (that is the derivative)

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.2I dunno if you simplified that correct for area. Shouldn't you get something like this? Remember the bottom r is squared. \[\huge A=2 \pi r^2+\frac{1000}{r}\]

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0yes that's what I took the derivative of

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.2Hmm, your second term looks a little off. We should get something like this, \[\huge A'=4\pi r\frac{1000}{r^2}\] Need to see steps?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0Nope, i made a mistake in the quotient rule

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0How do i solve for 'x' now?

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.2For r? Set equal to zero as you did. Then get a common denominator, turn it into one big fraction.

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0\[\huge 0=\frac{ 4(\pi r^3250) }{ \pi^2 }\] how do i solve for r now @zepdrix

dan815
 3 years ago
Best ResponseYou've already chosen the best response.0wut u mean its simple, solve for it, remember pi is just some constant

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.2\[\huge 0=\frac{ 4(\pi r^3250) }{ r^2 }\]Multiply both sides by r^2 giving us,\[\huge 0=4(\pi r^3250)\]Then divide both sides by 4, and solve! :)

dan815
 3 years ago
Best ResponseYou've already chosen the best response.0^ he means multiply by pi^2 but u get the point

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.2no, he put pi^2 on the bottom as a mistake.

dan815
 3 years ago
Best ResponseYou've already chosen the best response.0i was wondering why hed ask for help to solve that xD
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