Question 61016
a cylindrical tin of particular volume is to be made using as little material as possible. find the ratio of the height to the radius(the tin is closed both ends)
LET THE RATIO OF HEIGHT TO RADIUS = X =H/R.....H=RX
VOLUME OF CYLINDER = PI*(R^2)*H = V = CONSTANT.
V = PI*R^2*RX = PI*(R^3)*X
R=[V/(PI*X)]^(1/3)
A = AREA OF CYLINDER = 2PI*R*H+2PI*R^2=2PI*R[RX+R]=2PI*(R^2)[X+1]
A = 2PI*R^2[X+1]=2PI[X+1][V/(PI*X)]^(2/3)
A=[2PI*V^(2/3)/{PI}^(2/3)][(X+1)/X^(2/3)]
A = K[X^(1/3)+X^(-2/3)]
DA/DX = 0 FOR MINIMUM VALUE = K[(1/3){X^(-2/3)} -(2/3){X^(-5/3)}]=0
1/3{X^(2/3)}= 2/3{X^(5/3)}
X^(5/3)/X^(2/3)=2
X = 2