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Math Olympiad Test: Surface Area and Volume- 2 - Class 10 MCQ


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15 Questions MCQ Test Olympiad Preparation for Class 10 - Math Olympiad Test: Surface Area and Volume- 2

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Math Olympiad Test: Surface Area and Volume- 2 - Question 1

The aspherical shell of lead whose external and internal diameters are 24 cm and 18 cm respectively is melted and recast into a right circular cylinder 37 cm high. Find the diameter of the base of the cylinder.

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 1

The volume of lead in the shell = volume of a cylinder

⇒ r2 = 36 ⇒ r = 6
∴ diameter =  2r = 2 × 6 = 12 cm

Math Olympiad Test: Surface Area and Volume- 2 - Question 2

A cylindrical container is filled with icecream whose diameter is 12 cm and height 15 cm. The whole ice-cream cone?

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 2

Let r be the radius of base of the conical part
Height of the conical part = 4r
Volume of cone with hemispherical top = volume of cone + volume of hemispherical top

Volume of 10 such cones with hemispherical top
= 10 × 2πr3 = 20πr3 cm3
Volume of cylindrical container

20πr3 = 540π
r3 = 27 ⇒ r = 3 cm
diameter = 3  × 2 = 6 cm

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Math Olympiad Test: Surface Area and Volume- 2 - Question 3

A solid cylinder of diameter 12 cm and height 15 cm is melted and recast into 12 toys in the shape of a right circular cone mounted on a hemisphere. Find the radius of the hemisphere, if height of the cone is 3 times the radius.

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 3

Radius of cylinder = 12/2 = 6 cm
height = 15 cm
Volume of cylinder = πr2h
= 22/7 x 6x 15
= 540π cm
Volume of 12 toys = 540π cm3
Volume of 1 toy = 540π/12 = 45π cm3
Let the radius of the hemisphere be r cm.
Height of the cone = 3r cm
Volume of one toy = Volume of hemisphere + volume of cone

Now

Math Olympiad Test: Surface Area and Volume- 2 - Question 4

The volume of the cube is 1728 cm3. What is the total surface area of the cube?

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 4

The volume of the cube = 1728
a3 = 1728 ⇒ A = 12 cm
Total surface area = 6a2 = 6 × (12)2
= 6 × 144 = 864 cm2.

Math Olympiad Test: Surface Area and Volume- 2 - Question 5

A circus tent is cylindrical to a height of 4 cm and cone above it. If its diameter is 105 m and its lant height is 40 m. What is the total area of canvas required?

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 5

Area of canvas = 2πrh + πrl

= 1320 + 6600 = 7920 cm2.

Math Olympiad Test: Surface Area and Volume- 2 - Question 6

A right cylinder, a right cone and a hemisphere have the same height and same base area. What is ratio of their volumes?

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 6

Let the base areas and heights be A and h respectively.
For cylinder, volume = VC = πr2h = Ah
For cone, volume = VCO = 1/3 πr2h = 1/3 Ah
For hemisphere, volume = Vh = 2/3 πr3
∴ Ratio of their volumes = 2/3 (πr2)r
= 2/3 Ah [∵ h =r]

Math Olympiad Test: Surface Area and Volume- 2 - Question 7

The height of a right cylinder is 165 cm and the diameter of its base is 24 cm. Find the radius of the sphere whose volume is equal to the volume of the given cylinder.

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 7

Volume of given cylinder = πr2
= π(12)2 × 16
Now, 

Math Olympiad Test: Surface Area and Volume- 2 - Question 8

If the radius of a sphere is doubled then how many times will its surface area?

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 8

Surface area of sphere = 4πR2
If the radius of sphere is doubled, i.e., R becomes 2R, then
New surface area = 4π(2R)2 = 16πR2
= 4 (4πR2)
= 4 (surface area)
∴ Surface area will become 4 times.

Math Olympiad Test: Surface Area and Volume- 2 - Question 9

The inner and outer surface areas of a spherical shell are 324 πcm2 and 576 πcm2. What is the thickness of the shell?

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 9

Inner surface Area = 4πr2 = 324π cm2

∴ Outer radius

∴ thickness of the shell = (12 - 9) = 3 cm

Math Olympiad Test: Surface Area and Volume- 2 - Question 10

Five people will live in a tent. If each person requires 16 m2 of floor area and 100 m3 space for air then find the required height of the cone of the smallest size to accommodate those persons.

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 10

Let the height of the required cone be h cm
∴ Required base area = (16) × 5
= 80 cm2 = πr2
Height = h  cm volume = 1/3 (πr2)h
According to given condition
Total volume required = 5 × 100 cm3 = 500 cm3

Math Olympiad Test: Surface Area and Volume- 2 - Question 11

The ratio of the heights of two right cones is 3 : 2. Ratio of their radii of the base is 2 : 3. What is the ratio of their volumes?

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 11

Ratio of volumes 

Math Olympiad Test: Surface Area and Volume- 2 - Question 12

The total surface area of a right cone is 1760 cm2 and the radius of its base is 14 cm. What is the lateral surface area of the cone?

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 12

Total surface area of cylinder
= 2πr (r + h)
= 2 x 22/7 x 14(14 + h)
= 88 (14 + h)
= 1760 cm2.
⇒ h = 1760/88 - 14 = 6 cm
∴ Lateral Surface Area = 2 × π × r × h
= 2 x 22/7 x 14 x 6
= 498 cm2.

Math Olympiad Test: Surface Area and Volume- 2 - Question 13

On increasing each of the radius of the base and the height of a cone by 20%. By what percent its volume will be increased?

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 13

Let the original radius be r and height be h
Original volume = V = 1/3 πr2h
New radius = 120% of r = 120r/100 = 6r/5
New height = 120% of h = 120h/100 = 6h/5
New volume

Increase in volume 

Increase %

Math Olympiad Test: Surface Area and Volume- 2 - Question 14

The ratio between the volumes of two spheres is 8 : 27. What is the ratio between their surface area?

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 14

Let the radius of two spheres be R and r

Ratio of their surface area

Math Olympiad Test: Surface Area and Volume- 2 - Question 15

The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. What is the ratio of their volumes?

Detailed Solution for Math Olympiad Test: Surface Area and Volume- 2 - Question 15

Let the radii of two cylinders are 2r, 3r and heights be 5h and 3h
Ratio of their volumes

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