Solved Questions on Surface Area and Volume
1. The radius of the sphere, cone and cylinder is equal. Their total surface area is also equal. What is the ratio of their heights?
a) 1 : √2 : 1
b) 2 : 2 : 1
c) 2 : √2 : 1
d) 2 : 2√2 : 1
Answer: d) 2 : 2√2 : 1
Explanation: Let the radius of the sphere, cone and cylinder be R.
Let the height of the sphere be H1.
Let the height of the cone be H2.
Let the height of the cylinder be H3.
Let l be the slant height of the cone.
We know that the height of the sphere is the same as the diameter of the sphere.
H1 = 2R
We know that
Total Surface Area of the sphere = 4πR2
Total Surface Area of the cone = πR(l + R)
Total Surface Area of the cylinder = 2πR(H3 + R)
We are given that the total surface area of the sphere, cone and cylinder is equal.
So first taking,
Total surface area of the sphere = Total surface area of the cylinder
4πR2 = 2πR(H3 + R)
2R = H3 + R
R = H3
Now taking,
Total surface area of the sphere = Total surface area of the cone
4πR2 = πR(l + R)
4R = l + R
3R = l … (1)
In a cone,
(Slant height)2 = Height2 + Radius2
(l)2 = (H2)2 + (R)2
(3R)2 = (H2)2 + R2
9R2 = (H2)2 + R2
9R2 − R2 = (H2)2
8R2 = (H2)2
H2 = √8R2
H2 = 2√2 R
Now,
H1 : H2 : H3 = 2R : 2√2 R : R
= 2 : 2√2 : 1
2. A conical cup is filled with ice cream. The ice cream forms a hemispherical shape on its open top. The height of the hemispherical part is 4.2 cm. The height of the cone is equal to the radius of the hemispherical part. What is the volume of the ice cream?
a) 232.748 cm3
b) 232.648 cm3
c) 232.848 cm3
d) 232.548 cm3
Answer: c) 232.848 cm3
Explanation: We know that the volume of a hemisphere = 2πr3 / 3
Volume of a cone = πr2h / 3
We are given the height of the hemispherical part = 4.2 cm
Height of the hemispherical part = Radius of the hemisphere
The radius of the hemisphere = 4.2 cm
We are given that the height of the cone is equal to the radius of the hemispherical part.
Height of the cone = 4.2 cm
Now,
Volume of the ice cream = Volume of the hemispherical part + Volume of the cone
= 2πr3 / 3 + πr2h / 3
= [2⁄3 × 22⁄7 (4.2)3] + [1⁄3 × 22⁄7 × (4.2)2 × 4.2]
= [2⁄3 × 22⁄7 × 4.2 × 4.2 × 4.2] + [1⁄3 × 22⁄7 × (4.2)2 × 4.2]
= (2 × 22 × 1.4 × 0.6 × 4.2) + (22 × 1.4 × 0.6 × 4.2)
= (155.232) + (77.616)
= 232.848 cm3
3. If a solid metallic sphere of radius 9 cm is melted and recast into a right circular cone of base radius 4 cm, then what is the height of the cone?
a) 180.25 cm
b) 182.25 cm
c) 183.25 cm
d) 181.25 cm
Answer: b) 182.25 cm
Explanation: We know that the volume of a sphere = 4πr3 / 3
Volume of a cone = πr2h / 3
The radius of the metallic sphere = 9 cm
The radius of the cone = 4 cm
Now, Volume of the metallic sphere = 4π (9)3 / 3
= 4π (729) / 3
= 4π (243)
= 972π
Volume of the cone = π(4)2h / 3
= 16πh / 3
When one solid is melted and recast into another shape, its volume is the same.
→ Volume of the cone = Volume of the metallic sphere
→ 16πh / 3 = 972π
→ h = 972π×3⁄16π
→ h = 243×3⁄4
→ h = 729 / 4
→ h = 182.25 cm
4. If the area of the base of the frustum of a right circular cone is 25π cm2, the diameter of the circular upper surface is 6 cm and the slant height is 8 cm, then what will be the total surface area of the frustum?
a) 308 cm2
b) 307 cm2
c) 304 cm2
d) 306 cm2
Answer: a) 308 cm2
Explanation: Let the radius of the base of the frustum be R cm.
We are given that the area of the base of the frustum is 25π cm2.
We know that the area of the base is πr2
πR2 = 25π
R2 = 25
R = √25
R = 5 cm
We are given the diameter of the circular upper surface = 6 cm
The radius of the circular upper surface (r) = 3 cm
We are given that the slant height (l) = 8 cm
We know that the total surface area of the frustum = πR2 + πr2 + πl(R + r)
→ Total surface area of frustum = 22⁄7 × (5)2 + 22⁄7 × (3)2 + 22⁄7 × 8 × (5 + 3)
= 22⁄7 × 25 + 22⁄7 × 9 + 22⁄7 × 8 × 8
= 22⁄7 × (25 + 9 + 64)
= 22⁄7 × 98
= 22 × 14
= 308 cm2
5. If a solid sphere of radius 7 cm is melted and moulded into small identical cubes of side length 2 cm, then how many such cubes can be formed from the sphere?
a) 178
b) 181
c) 180
d) 179
Answer: d) 179
Explanation: We know that the volume of a sphere = 4πr3 / 3
Volume of the cube = (side)3
Volume of the sphere = 4πr3 / 3
= 4π(7)3 / 3
= (4 × (22 / 7) × (7)3 / 3
= (4 × 22 × (7)2 / 3
= 4312 / 3
= 1437.33 cm3
Volume of the cube = (side)3
= (2)3
= 8 cm3
When one solid is melted and recast into another shape, its volume is the same.
Thus, Number of cubes formed = (Volume of the sphere) / (Volume of the cube)
= 1437.33 / 8
= 179.67
~ 179
Thus, 179 identical cubes of side 2 cm can be formed from the solid sphere with a radius of 7 cm when melted and moulded.
