# A given quanitity of metal is to be cast into a half cylinder with a rectangular base and semi circular ends. Finfd ratio of length of the cylinder to the diameter of its semi-circular ends.

A given quantity of metal is to be cast into a half cylinder with a rectangular base and semi circular ends.in order to show that total surface area may be minimum. Find ratio of length of the cylinder to the diameter of its semi-circular ends

let h be the height of the cylinder and r be the radius of the semicircular ends.

let V be the volume of the half cylinder, therefore

.............(1) here we have V is constant.

let S be the total surface area of the half cylinder. then

S = area of the rectangular base + area of the two semi circular ends + area of the curved surface

...................(2)

since V is constant, substituting the value of h in terms of V from (1) i.e.

.............(3)

differentiating eq (3) wrt r :

...............(4)

now differentiating (4) wrt r:

...............(5)

for S to be minimum equating dS/dr = 0

for this value of r , is positive, thus for this value S is minimum

therefore

which is the required ratio of the length of the half cylinder to the diameter of its semi circular ends.

hope this helps you.

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