Question

/ 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.​

Answer 1

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.