A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet. If the capacity of the jar is to be maximum, then the diameter of the jar must be k times the height of the jar. The value of k is???
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Answer:
the diameter be D and the height be H.
The surface area S of the metal sheet, which is constant, can be expressed as the sum of the lateral surface area and the base of the cylinder.
S=πDH+π
4
D
2
Hence.
H=
πD
1
(S−
4
πD
2
)
=
πD
S
−
4
D
The capacity, or volume, V of the cylinder is given as:
V=π
4
D
2
H
=π
4
D
2
(
πD
S
−
4
D
)
=
4
DS
−
16
πD
3
)
For the maxmium volume,
dD
dV
=0
∴
4
S
−
16
3πD
2
=0
S=
4
3πD
2
We showed earlier that
H=
πD
S
−
4
D
∴H=
4
3D
−
4
D
=
2
D
Hence, the diameter D is 2 times the height.
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