SOLUTION: 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)

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Question 61016This question is from textbook as pure mathematics
: 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) This question is from textbook as pure mathematics

Answer by venugopalramana(3286) (Show Source):
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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