Question

A cylindrical can is to be made to hold 1500 cubic inches of oil. Find the radius
and height of the can so as to minimize the cost of material to manufacture the can.

Answer 1

\[V_\text{cylinder}=\pi r^2h\]
there are two variables and two equations will be required.
Where is the information of the "cost"??

Answer 2

thats all the information it gives

Answer 3

You need to minimize surface area (this will minimize material cost), while having the volume as 1500 cubic inches.

\[Surface Area = 2 \pi r^2 + 2 \pi r h \]

Answer 4

@agent0smith but where does it say that the relation of cost given?

Answer 5

\[1500= \pi r^2 h\]
You can use this to solve for h or r, to put into the SA equation, then differentiate SA and let it equal zero, and solve for h (or r).

Answer 6

@electrokid it doesn't matter, by minimizing SA, we minimize total material and thus material costs. It's a can, so the top and sides are both made of the same material.

Answer 7

@agent0smith so much for the implicit questioners

Answer 8

Yeah, we kinda have to just assume that the top/bottom and sides are made of the same material and cost the same to make.

Answer 9

im sorry im still kind of confused as to how i will set up this equation. when the formula 1500=πr2h requires that i have both radius and height

Answer 10

haha you're in my math class but yeah, this one doesn't make sense because it seems like we need either r or h to find the equations and we were only given the volume

Answer 11

which ones have you done?

Answer 12

I've only gotten through 1-6 and I'm on 8 now because I'm skipping 7

Answer 13

I think 7 goes something like \[h=2500\div(\pi r^2)\] and then you plug that in for h into the equation \[2\pi r^2 + 2\pi r (1000/\pi r^2) \] and then you differentiate and set it to 0 but I don't know what the derivative is because I sucked at learning all the rules

Answer 14

@nickaayy is on the right track. You rearrange 1500=πr2h to get h=... or r=... and then sub that into the surface area equation, and differentiate, and set it to zero.

Answer 15

\[1500= \pi r^2 h\]so \[h = \frac{ 1500 }{ \pi r^2 }\]
Now sub that in for h in the SA equation: \[SA = 2 \pi r^2 + 2 \pi r h\] to get \[SA = 2 \pi r^2 + 2 \pi r \left( \frac{ 1500 }{ \pi r^2 }\right) \] then simplify a bit to get: \[\large SA = 2 \pi r^2 + 3000 \pi r ^{-1}\] and differentiate, set to equal zero.