SURFACE AREAS
AND VOLUMES
Q. A cylindrical bucket, 32 cm high and with radius of base
18 cm, is filled with sand. This bucket is emptied on the ground
and a conical heap of sand is formed. If the height of the conical
heap is 24 cm, find the radius and slant height of the heap.
h1 = 32cm
h2 = 24cm
=
Vol. of cylindrical bucket (V1)
Vol. of cone (V2)
π r12 h1
πr22h2
×
1
3
?
✔
✔
✔
Sol.
What is the formula to find volume of cylinder?
What is the formula to find volume of a cone?
The cylindrical bucket filled with sand is emptied on the ground and a conical heap of sand is formed.
l = ?
r2 = ?
r1 = 18 cm
What is the relation between the volume of the cylindrical bucket and volume of cone?
They are equal
=
Vol. of cylindrical bucket (V1)
Vol. of cone (V2)
π
×
r12
×
h1
=
∴
Q. A cylindrical bucket, 32 cm high and with radius of base
18 cm, is filled with sand. This bucket is emptied on the ground
and a conical heap of sand is formed. If the height of the conical
heap is 24 cm, find the radius and slant height of the heap.
Sol.
Vol. of cylindrical bucket
=
Vol. of the cone
×
π
×
r22
×
h2
1
3
×
r22
=
18
×
18
×
32
×
24
∴
1
3
r22
=
3
8
4
18 × 18 × 32 × 3
24
r22
=
3
×
18
×
8
r22
=
3
×
3 × 3 × 2
×
2
r2
=
3
×
3
×
2
×
2
×
3
×
3
∴
∴
∴
∴
π r12 h1
πr22h2
×
1
3
?
h2 = 24cm
l = ?
r2 = ?
r1 = 18 cm
h1 = 32cm
2
×
× 2
h2 = 24cm
l = ?
r2 =
r1 = 18 cm
h1 = 32cm
=
Vol. of cylindrical bucket (V1)
Vol. of cone (V2)
Q. A cylindrical bucket, 32 cm high and with radius of base
18 cm, is filled with sand. This bucket is emptied on the ground
and a conical heap of sand is formed. If the height of the conical
heap is 24 cm, find the radius and slant height of the heap.
Sol.
r2
=
3
×
3
×
2
×
2
r2
=
36 cm
∴ Radius of the heap is 36 cm
Now,
l2
=
r2
+
h2
∴
l2
=
(36)2
+
(24)2
∴
l2
=
1296
+
576
∴
l2
=
1872
∴
l
=
What is the formula to find slant height (l ) ?
l2
=
r2
+
h2
2
1872
936
2
468
2
234
2
117
3
39
3
13
13
1
∴
l
=
π r12 h1
πr22h2
×
1
3
?
✔
∴
Now, let us find slant
height of the cone
∴
∴
?
36 cm
=
Vol. of cylindrical bucket (V1)
Vol. of cone (V2)
Q. A cylindrical bucket, 32 cm high and with radius of base
18 cm, is filled with sand. This bucket is emptied on the ground
and a conical heap of sand is formed. If the height of the conical
heap is 24 cm, find the radius and slant height of the heap.
Sol.
l
=
∴
l
=
=
2
× 2
× 3
l
∴
h2 = 24cm
l =
r2 = ?
r1 = 18 cm
h1 = 32cm
?