VOLUME OF CONE QUESTIONS AND ANSWERS

Question 1 :

Radius and slant height of a cone are 20 cm and 29 cm respectively. Find its volume.

Solution :

Radius of the cone (r)  =  20 cm

Slant height of the cone (l)  =  29 cm

l²  =  r²+h²

29²  =  20² + h²

841 = 400 + h²

h²  =  841-400

 h²  =  441

 h  =  √(21 21

 h  =  21 cm

Volume of the cone  =  (1/3) Π r² h

=   (1/3) ⋅ (22/7) ⋅ (20)² ⋅ 21

=  8800 cm³ 

Volume of the cone = 8800 cm³

Question 2 :

The circumference of the base of a 12 m high wooden solid cone is 44 m. Find the volume.

Solution :

Circumference of cone  =  44 m

Height of the cone (h)  =  12 m

2Πr  =  44

2 ⋅ (22/7) ⋅ r  =  44

r  =  44 ⋅ (1/2) ⋅ (7/22)

r  =  7 cm

Volume of the cone  =  (1/3) Π r² h

=  (1/3) ⋅ (22/7) ⋅ 7² ⋅ 12

=  (1/3) ⋅ (22/7) ⋅ 7 ⋅ 7 ⋅ 12

=  616 cm³ 

Volume of the cone  =  616 cm³ 

Question 3 :

A vessel is in the form of frustum of a cone. Its radius at one end and the height are 8 cm and 14 cm respectively. If its volume is 5676/3 cm³, then find the radius at the other end.

Solution :

Volume of the frustum cone  =  (5676/3) cm³

Let r be the required radius

Radius (R)  =  8 cm

height (h)  =  14 cm

(1/3) Π h (R²+r²+R r) = (5676/3)

(1/3) ⋅ (22/7) ⋅ (14) (8²+ r²+8r) = 5676/3

r²+8r+64  =  129

r²+ 8r+64-29  =  0

r²+8r-65  =  0

(r+13) (r-5)  =  0

r  =  -13, r  =  5 cm

So, the required radius = 5 cm

Question 4 :

The perimeter of the ends of a frustum of a cone are 44 cm and 8.4 Π cm. If the depth is 14 cm, then find its volume.

Solution :

Perimeter of upper end  =  44 cm

Perimeter of lower end  =  8.4 Π cm

Height of frustum cone  =  14 cm

Now we have to find the volume of frustum cone

Volume of the frustum cone  =  (1/3) Π h (R²+r²+R r)

2ΠR  =  44 

2 ⋅ (22/7) ⋅ R  =  44      

R  =  44 ⋅ (1/2) ⋅ (7/22)

R  =  2 ⋅ (1/2) ⋅ 7

R  =  7

2Πr  =  8.4 Π

r  =  8.4 Π ⋅ (1/2Π)

r  =  4.2 

Volume of the frustum cone

=  (1/3) ⋅ (22/7) ⋅ 14 (7²+4.2²+7(4.2))

=  (44/3) (49+29.4+17.64)

=  (44/3) (96.04)

=  (44) (32.013)

=  1408.57 cm³

Volume of the frustum cone =  1408.57 cm³

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