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.
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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.
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