/ The area
of curved surface of right
circular cone is
root over 5 times that of the base of the cone, find the ratio of
the height and the radius of the base of the Cone.
Answers
Answered by
0
Step-by-step explanation:
Let, r,l and h be the radius, slant height and altitude of the cone respectively, then
V=
3
1
πr
2
h⟶(1)
& h=
πr
2
3V
⟶(2)
S=πrl, squaring both sides
S
2
=π
2
r
2
l
2
=πr
2
(h
2
+r
2
)[∵l
2
=r
2
+h
2
]
=π
2
r
2
[
π
2
r
4
9V
2
+r
2
][from(2)]
S
2
=
r
2
9V
2
+π
2
r
4
, differentiating with respect to
′
r
′
2S
dr
ds
=
r
3
−18V
2
+4π
2
r
3
For maximum or minimum,
dr
ds
=0⇒
r
3
−18V
2
+4π
2
r
3
=0
⇒4π
2
r
3
=
r
3
18V
2
⇒2π
2
r
6
=9V
2
=9×
9
1
π
2
r
4
h
2
[from(1)]
⇒2π
2
r
6
=π
2
r
4
h
2
⇒2r
2
=h
2
⇒h=
2
r
dr
2
d
2
S
=12π
2
r
2
+
r
4
54V
2
>0 for all values of V and r.
Hence, for the least surface area of a cone and given volume, altitude is equal to
2
times the radius of the base.
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