cylinder: minimal surfice area w/ volume of 1 liter

[Math wiz ya rite 09]

You are designing a new cylindrical can to hold 1 liter (1000 cubic centimeters) of Jake's Special Paint. Find the dimensions that will minimize the cost of metal to manufacture the can.

 

[daon]

Volume of cylinder:
$\displaystyle V\, =\, 1000cm^3\, =\, \pi r^2h$

Surface area of cylinder: Top+Bottom+Shaft

$\displaystyle A\, =\, 2(\pi r^2)\, +\, 2\pi r h$

You want to minimize the surace area, so solve for either r or h in the volume formula (I will demonstrate solving for h).

$\displaystyle h\, =\, \frac{1000cm^3}{\pi r^2}$

Then...

$\displaystyle A\, =\, 2(\pi r^2)\, +\, 2\pi r \frac{1000cm^3}{\pi r^2}\, =\, 2 \pi r^2\, +\, \frac{2000cm^3}{r}$

Notice A is a function of r now. You can take the derivative of A with respect to r and set equal to zero to get an r that will give you a minimum area. Use that r in the volume equation to get the height.

 

[mathwizz420]

If the volume of the can was simply any V cubic units how would you should that the minimum surface area is achieved when the height is equal to the diameter of the base or 2R

 

[mathwizz420]

I have already found how to show that the height is equal to two times the radius but Im not sure how to find the minimum surface are using this. When I tried to solve it out I got to r=(V/Pi)^(1/3)...am i on the right path

 

[Deleted member 4993]

I have already found how to show that the height is equal to two times the radius but Im not sure how to find the minimum surface are using this. When I tried to solve it out I got to r=(V/Pi)^(1/3)...am i on the right path

correct