MCQ: Hemispheres - SSC CGL MCQ

Given:

Height of the tent = 22.5 m
the diameters of the upper and the lower circular ends of the frustum are 21 m and 39 m, respectively

Formula used:

Slant height of a frustum (l) = √[H² + (R - r)²]
Curved surface area of a frustum = πl(R + r)
Curved surface area of a hemisphere = 2πr2
Here,
H = Height of the frustum
R = Lower radius
r = upper radius and the radius of the hemisphere

Calculation:

In the given figure,
Height of the frustum = 12 m and height of the hemisphere = 10.5 m
Upper radius = 21/2
Lower radius = 39/2
Now,
Slant height (l) of the frustum = √[12² + {(39/2) - (21/2)}²]

Surface area of the tent = Curved surface area of the frustum + curved surface area of the hemisphere
So,
Surface area of the tent = 15π(39/2 + 21/2) + 2π × (21/2)²

So, the required area of the cloth 

∴ The area of the cloth (in m2) used to make the tent is 

Given:

Outer diameter of rubber ball = 22 inches.

Thickenss = 0.5 inches

Formula used:

Surface area of sphere = 4πr² [where, r is the radius of sphere]

Inner radius of sphere = outer radius - thickness

Diameter = 2 × radius

Calculations:

Outer diameter = 22

Inner radius of rubber ball = 11 - 0.5 = 10.5 inches
Now, surface area of inner side of rubber ball = 4 x 3.14 x 10.52
= 4 × 3.14 × 110.25
= 1384.74 sq. inches
∴ The answer is 1384.74 sq. inches.

Given:

Total surface area of hemisphere = 462 cm²

Formula:

Total surface area of hemisphere = 3πr²

Calculation:

According to the question

Total surface area of hemisphere = 462

∴ Diameter of hemisphere = 7 × 2 = 14 cm

Given:

Inner Diameter = 84 cm
Cost of tin plating it on the inside = Rs. 21 per cm².

Concept Used:

Curved Surface area of Hemisphere = 2π r²

Calculation:

Inner Diameter = 84 cm
Inner Radius = 84/2 cm = 42 cm
Curved Surface area of Hemisphere = 2π r²

The cost of tin-plating 100 cm² of the bowl = Rs. 21
The cost of tin-plating 1 cm² of the bowl = Rs. 21/100
The cost of tin-plating 11088 cm² area of the bowl = (21/100) × 11088 = Rs. 2,328.48
Thus, the cost of tin-plating is Rs. 2,328.48.
∴ Option 1 is the correct answer.

Given:

Radius of the hemisphere = 3.5 cm

Formula used:

Volume of a hemisphere = (2/3)πr³
Volume of a sphere = (4/3)πr³
r = Radius

Calculation:
Let the radius of spheres be r
According to the question:

∴ Radius of small spheres = 7/4 cm

Given:

A cone, a hemisphere and a cylinder stand on equal bases and have same height.

Formula used:

Volume of cone = 1/3 πr²h

Volume of hemisphere = 2/3πr³

Volume of cylinder = πr²h

Calculation:

A cone, a hemisphere and a cylinder stand on equal bases and have same height.

Hence radius of base of cone = radius of hemisphere = radius of cylinder = r
They also have same height = h
Height of cone = h
Height of cylinder = h
Height of hemisphere = r = h
∴ Volume of cone = V₁= 1/3π r²h
Volume of cylinder = V₂ = πr2h
Volume of hemisphere = V3 = 2/3π r² × r
⇒ 2/3 π r² × h = 2/3π r²h
Ratio of volume of cone : hemisphere : cylinder 

⇒  1/3π r²h :  2/3π r²h :  πr²h
⇒ 1/3 : 2/3 : 1
⇒ 1 : 2 : 3
∴ Option 1 is correct.

Given:

The radius of a solid hemisphere is 21 cm.
The ratio of the cylinder's curved surface area to its Total surface area is 2/5.

Formula used:

The curved surface area of the cylinder = 2πRh
The total surface area of cylinder = 2πR(R + h)
The volume of the cylinder = πR²h
The volume of the solid hemisphere = 2/3πr³ 
(where r is the radius of a solid hemisphere and R is the radius of a cylinder)

Calculations:
According to the question,
CSA/TSA = 2/5
⇒ [2πRh]/[2πR(R + h)] = 2/5
⇒ h/(R + h) = 2/5

⇒ 5h = 2R + 2h

⇒ h = (2/3)R .......(1)

The cylinder's volume and the volume of a solid hemisphere are equal.

⇒ πR²h = (2/3)πr³
⇒ R² × (2/3)R = (2/3) × (21)³
⇒ R³ = (21)³
⇒ R = 21 cm
∴ The radius (in cm) of its base is 21 cm.

Given: 

The internal radius of the tank is 10.5 m
A pipe at the rate of 7.7 liters per second emptied the tank.

Concept:  
Volume of hemisphere = 2π/3 x r³
1 m³ = 1000 L
1000 cm³ = 1 L

Calculation: 
The volume of the hemispherical tank is 
⇒ 2/3 × 22/7 x 10.5 x 10.5 x 10.5
⇒ 2425.5 m³

The capacity of the tank is 
⇒ 2425.5 × 1000 L
⇒ 2425500 L
So, 
Time taken by the pipe emptied 2/3 part of the tank is 
⇒ (2/3 x 2425500) ÷ 7.7 sec
⇒ 210,000 sec
Time in hours 
⇒ 210,000/3600
⇒ 175/3 hours 
∴ The required time is 175/3 hours.

Given:

Diameter of the hemispherical depression = 4 cm
Each face of the cube = 10 cm

Concept used:

TSA of the cube = 6a²
CSA of the hemisphere = 2πr²
Area of a circle = πr²
a = side or edge of a cube
r = radius of circle or hemisphere

Calculation:

The figure below of the solid is created as per the given information along with the top view of the solid,

This hemispherical shape will be formed on each face of the cube

Radius of the hemisphere = 4/2 = 2 cm
According to the question,
Required surface area = TSA of the cube - 6 × areas of circle form on the top of each face of the cube + 6 × CSA of hemisphere formed on each face of a cube
⇒ Required surface area = 6 x 10² - 6 x π2² + 6 x 2π2²
⇒ Required surface area = 600 + 6π2²
⇒ Required surface area = 600 + 24 x 22/7
⇒ Required surface area = 600 + 528/7

∴ The surface area of the remaining solid (in cm²) is  

Given:

The diameter of a hemisphere is 28 cm.

Concept used:

1. Volume of a hemisphere = 2πR³/3 (R = Radius)
2. Diameter = Radius × 2

Calculation:

Radius of the hemisphere = 28/2 = 14 cm
Now, the volume of the hemisphere

∴ The volume of the hemisphere is 5749.33 cm³.

Given:

The volume of a solid hemisphere is  
Take π = 22/7

Formula used:

The volume of the solid hemisphere = 2/3πr³
The total surface area of the solid hemisphere = 3πr²

Where, 
r, is the radius of the hemisphere

Calculation:

According to the question, the required figure is:

The volume of the solid hemisphere,

Now, 

The total surface area of the solid hemisphere  

∴ The total surface area of the solid hemisphere is 594/7 cm²

Given:

Outer radius (R) of hemispherical bowl = 45 cm.
Inner radius (r) of hemispherical bowl = 41 cm.

Formula used:

Total surface area (T.S.A) of hemispherical shell 

Calculation:

Using the above formula we get:

∴Total surface area of the bowl is 7756π cm². 

Given:

TSA of Hemisphere = 16632 cm²

Formula used:

Volume of solid hemisphere 
 
Calculation:-

It is given that the total surface area of the hemisphere is 16632 cm.

Total surface area = 3 π r²

∴ The volume of the solid hemisphere is 155232 cm³.

Given::

Inner radius = 6 cm
Thickness = 0.37 cm

Formula used:

Outer radius = Inner radius + Thickness
Curved surface area of the hemisphere = 2πr²

Calculations:

According to the formula,
⇒ Outer radius = 6  + 0.37 = 6.37 cm

⇒ Hence, Curved surface area of the hemisphere is 255.0548 cm²

Given:

Radius of large hemisphere = 3 × radius of small hemisphere

Concepts used:

Volume of hemisphere = (2/3) × π × (radius)³
Ratio = [(Volume of large hemisphere – volume of small hemisphere)/Volume of small hemisphere]

Calculation:

Let radius of small hemisphere be r units.
⇒ Radius of large hemisphere = 3r
Volume of small hemisphere = 2πr³/3 units
⇒ Volume of large hemisphere = 2π/3 × (3r)³ units = 18πr³ units
Ratio = [{18πr³ – (2πr³/3)}/2πr3/3] = 26 ∶ 1
∴ Ratio of difference between the volume of large hemisphere and the volume of small hemisphere to the volume of smaller hemisphere is 26 ∶ 1.