Question

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.

Answer 1

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):\[\large A=2\pi r^2+2 \pi r h\]

Answer 2

They want us to minimize the surface area, given a constraint on the volume.

Answer 3

\[\large 500=\pi r^2h\]Solving for h gives us,\[\large \color{orangered}{h=\frac{500}{\pi r^2}}\]

We'll plug this into our Area formula.
\[\large A=2\pi r^2+2 \pi r \color{orangered}{h} \qquad\rightarrow\qquad A=2\pi r^2+2 \pi r \color{orangered}{\frac{500}{\pi r^2}}\]

Answer 4

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.

Answer 5

\[\huge A=2 \pi r^2+1000\]

Answer 6

\[\huge 0=4\pi r-1000r\]

Answer 7

am i doing it right so far (that is the derivative)

Answer 8

I dunno if you simplified that correct for area.
Shouldn't you get something like this?
Remember the bottom r is squared.

\[\huge A=2 \pi r^2+\frac{1000}{r}\]

Answer 9

yes that's what I took the derivative of

Answer 10

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?

Answer 11

Nope, i made a mistake in the quotient rule

Answer 12

How do i solve for 'x' now?

Answer 13

For r?
Set equal to zero as you did.
Then get a common denominator, turn it into one big fraction.

Answer 14

oh yes okay !

Answer 15

\[\huge 0=\frac{ 4(\pi r^3-250) }{ \pi^2 }\]
how do i solve for r now @zepdrix

Answer 16

wut u mean its simple, solve for it, remember pi is just some constant

Answer 17

\[\huge 0=\frac{ 4(\pi r^3-250) }{ r^2 }\]Multiply both sides by r^2 giving us,\[\huge 0=4(\pi r^3-250)\]Then divide both sides by 4, and solve! :)

Answer 18

^ he means multiply by pi^2 but u get the point

Answer 19

no, he put pi^2 on the bottom as a mistake.

Answer 20

oh i see

Answer 21

i was wondering why hed ask for help to solve that xD

Answer 22

heh