The perimeter of the rectangle with length L and breadth b is 2(l + b)
Therefore,
2(L + b) = 36
L + b = 18
b = 18 - L
Let the rectangle be rotated about its breadth.
Then the resulting cylinder formed will be of radius L and height b.
Volume of cylinder formed V = πL2b = π(18L2 - L3)
To find the dimensions that will result in the maximum volume :
\(\frac{dV}{dL}\) = π(18 x 2 x L - 3 x (L2) = 0
36L = 3 x (L2)
L = 12,0
L cannot be 0. L is taken as 12 cm.
Therefore,
b = 24.
\(\frac{d^2V}{dL^2}\) = π(18 x 2 - 3 x (L))
At L = 12,
\(\frac{d^2V}{dL^2}\) = - 36π
= a -ve value
Therefore a maxima exists at L = 12, meaning the volume of the constructed cylinder will be maximum at L = 12 cm.