a company is manufacturer of cylinrical aluminium tins.the manager plans 2 reduce d cost.?

Question

the cost is proportional 2 d area of d aluminium sheet.d volume tat each tin can hold is 1000cm3 (1liter)....tat my project work...waiting a genius 2 help me!!!
1)determine d value of h,r and hence calculate d ratio of h/r when d total surface area of each tin is minimum.here,hcm denotes d height n rcm d radius of d tin.

Answer 1

The surface area of a cylinder with radius r and height h is 2*pi*r*h+2*pi*r^2, and the volume is pi*r^2*h. I'm assuming the tin must have both a top and a bottom.

Since the volume is 1000, you know that h=1000/(pi*r^2), so the surface area is 2*pi*r*(1000/(pi*r^2)) + 2*pi*r^2 = 2000/r + 2*pi*r^2.

To minimize the surface area, differentiate the surface area with respect to r, and you get

-2000/r^2 + 4*pi*r.

We want to find the value of r for which this is zero. Set it equal to zero and multiply both sides by r^2:

-2000 + 4*pi*r^3 = 0

Rearrange:

4*pi*r^3 = 2000

Solve for r:

r = cbrt(500/pi) = 5.42 cm.

Since h=1000/(pi*r^2), you have h = 10.84 cm. So the ratio h/r is 2:1.