design a cylinder which can hold a volume of 10ft cubed. so that it's used with minimal material. what are the dimensions (r and h) and what is the relationship between r and h.HINT:MINIMIZE SURFACE AREA????????????

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- Nov 23rd 2008, 03:15 PMla_lo626I need to minimize the surface area of this cylinder
design a cylinder which can hold a volume of 10ft cubed. so that it's used with minimal material. what are the dimensions (r and h) and what is the relationship between r and h.

**HINT:**MINIMIZE SURFACE AREA???????????? - Nov 23rd 2008, 04:15 PMTheEmptySet
Here is the basic set up.

The Volume of the cylinder is

If the cylinder is closed (like a can of soup) the surface area will be the area of the circles on the top and bottom and the area of the side of the cylinder.

if you cut open a cylinder you would get two circles and one rectangle.

Now The big question is how long is the rectangle? It's length must be the same as the circumference of the circle. So its area will be

and the area of the circle is

We can use these to find the surface area of the can.

I'm not sure if your cylinder is open on top or not, but the process for both is similar.

So lets suppose that it is open on top then the surface area is

but we also know that

Now we can sub this into the surface area equation to get

You should be able to finish from here.

Good luck - Nov 23rd 2008, 07:58 PMSoroban
Hello, la_lo626!

Quote:

Design a cylinder which can hold a volume of 10 ft³

so that its surface area is a minimum.

(a)What are the dimensions, and ?

The surface area is: . .[2]

Substitute [1] into [2]: .

Differentiate and equation to zero: .

Multiply by

Quote:

(b) What is the relationship between and ?

Ha . . . beat me to it . . .

.