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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.
 9 months ago
 9 months 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.
 9 months ago
 9 months ago

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zepdrixBest ResponseYou've already chosen the best response.2
Volume 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):\[\large A=2\pi r^2+2 \pi r h\]
 9 months ago

zepdrixBest ResponseYou've already chosen the best response.2
They want us to minimize the surface area, given a constraint on the volume.
 9 months ago

zepdrixBest 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. \[\large A=2\pi r^2+2 \pi r \color{orangered}{h} \qquad\rightarrow\qquad A=2\pi r^2+2 \pi r \color{orangered}{\frac{500}{\pi r^2}}\]
 9 months ago

zepdrixBest ResponseYou've already chosen the best response.2
Simplify 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.
 9 months ago

burhan101Best ResponseYou've already chosen the best response.0
\[\huge A=2 \pi r^2+1000\]
 9 months ago

burhan101Best ResponseYou've already chosen the best response.0
\[\huge 0=4\pi r1000r\]
 9 months ago

burhan101Best ResponseYou've already chosen the best response.0
am i doing it right so far (that is the derivative)
 9 months ago

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

burhan101Best ResponseYou've already chosen the best response.0
yes that's what I took the derivative of
 9 months ago

zepdrixBest ResponseYou've already chosen the best response.2
Hmm, 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?
 9 months ago

burhan101Best ResponseYou've already chosen the best response.0
Nope, i made a mistake in the quotient rule
 9 months ago

burhan101Best ResponseYou've already chosen the best response.0
How do i solve for 'x' now?
 9 months ago

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

burhan101Best 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
 9 months ago

dan815Best ResponseYou've already chosen the best response.0
wut u mean its simple, solve for it, remember pi is just some constant
 9 months ago

zepdrixBest 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! :)
 9 months ago

dan815Best ResponseYou've already chosen the best response.0
^ he means multiply by pi^2 but u get the point
 9 months ago

zepdrixBest ResponseYou've already chosen the best response.2
no, he put pi^2 on the bottom as a mistake.
 9 months ago

dan815Best ResponseYou've already chosen the best response.0
i was wondering why hed ask for help to solve that xD
 9 months ago
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