a cylindrical shaped tin can must have a capacity of 1000cm^3
determine the dimensions that require the minimum amount of tin for the can. (assume no waste material) according to the marketing department, the smallest can that the market will accept has a diameter of 6 cm and a height of 4 cm
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For a given volume, the smallest surface area of a cylinder is obtained when its height is equal to its diameter.
The surface area (S) of a cylinder in terms of its diameter (D) and volume (V) is given by the following equation.
S = (πD2/2) + (4V/D)
The 1st derivative of this with respect to D (at a given V) is
dS/dD = πD - (4V/D2)
Replacing V in the 2nd term by its equivalent (πHD2/4), where H is the cylinder's height, gives
dS/dD = πD - πH
At a minimum, dS/dD must be zero, which can happen only if H = D. It follows that for a given volume, the smallest surface area of a cylinder is obtained when its height is equal to its diameter, or when the height to diameter ratio is one.