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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