PROBLEMS ON SURFACE AREA OF SPHERE AND HEMISPHERE

Problem 1 :

If the curved surface area of solid sphere is 98.56 cm², then find the radius of the sphere.

Solution :

Curved surface area of sphere  =  98.56 cm²

4 Π r²  =  98.56

4 ⋅ (22/7) ⋅ r²  = 98.56

r²  =  98.56 ⋅ (1/4) ⋅ (7/22)

r²  =  98.56 ⋅ (1/4) ⋅ (7/22)

r²  =  7.84

r  =  √(2.8 ⋅ 2.8) 

r  =  2.8 cm

So, radius of the sphere is 2.8 cm.

Problem 2 :

If the curved surface area of the solid hemisphere is 2772 sq.cm, then find its total surface area.

Solution  :

Curved surface area of hemisphere  =  2772 cm²

2Πr²  =  2772

2 ⋅ (22/7) ⋅ r²  =  2772

r²  =  2772 ⋅ (1/2) ⋅ (7/22)

r²  =  441

r  =  21

Total surface area of hemisphere  =  3Πr²

=  3 ⋅ (22/7) ⋅(21)²

=  4158 cm² 

Total surface area of sphere = 4158 cm²  

Problem 3 :

Radii of two solid hemispheres are in the ratio 3:5. Find the ratio of their curved surface areas and the ratio of their total surface areas.

Solution :

Let r₁ and r₂ are the radii of two hemispheres

r₁ : r₂  =  3:5

r₁ / r₂  =  3/5

r₁  =  3r₂/5

Curved surface area of hemisphere  =  2Πr²

Ratio of curved surface area of two hemisphere

2 Π r² : 2 Π r²

(3 r₂/5)² : r₂²

9 : 25

Total surface area of hemisphere =  3Πr²

Ratio of curved surface area of two hemisphere

3Π r₁² : 3 Πr₂²

(3 r₂/5)² : r₂²

9 : 25

Ratio of curved surface area is 9 : 25

Ratio of total surface area is 9 : 25

Problem 4 :

A hemispherical bowl has a radius of 15 cm. If it is filled completely with water and covered with a lid,

(a) find the volume of the water

(b) find the surface area of the bowl (including the lid). Give your answers to 3 significant figures.

Take π = 3.142

Solution :

Radius = 15 cm

a) Volume of hemisphere = (2/3) Πr³

= (2/3) x 3.14 x 15³

= 7065 cm³

So, the required volume is 7065 cm³.

b) Surface area bowl = 4Πr²

= 4 x 3.14 x 15² 

= 39.44 cm²

So, the required surface area is 39.44 cm²

Problem 5 :

A Puffer fish is able to “puff up” when threatened by gulping water and inflating its body. The puffer fish is approximately a sphere with a diameter of 5 inches. Its surface area when inflated is about 1.5 times its normal surface area.

What is the surface area of the fish when it is not puffed up?

Solution :

Surface area of sphere = 4Πr²

Diameter = 5 inches, radius = 5/2 ==> 2.5 inches

Surface area of puffer fish after puff up :

= 4 x 3.14 x (2.5)²

= 78.5 cm²

(Surface area of puffer fish before puff up) x 1.5 = 78.5

Surface area of puffer fish before puff up = 78.5/1.5

= 52.3 cm²

Problem 6 :

A sphere has radius 4.5 cm. Find the volume and surface area of the sphere. Give your answers to 3 significant figures.

Take π = 3.142

Solution :

Radius of sphere = 4.5 cm

Volume of sphere = (4/3) Πr³

= (4/3) x 3.14 x (4.5)³

= (4/3) x 3.14 x 91.12

= 381.48 cm³

So, the required volume of the sphere is 381.48 cm³

Surface area of sphere = 4Πr²

= 4 x 3.14 x (4.5)²

= 254.34 cm²

So, the surface area of the sphere is 254.34  cm²

Problem 7 :

Basketballs used in professional games must have a circumference of 29 1/2 inches. What is the surface area of a basketball used in a professional game?

Solution :

Circumference of the circle = 2Πr

2Πr = 29  1/2 inches

2 x 3.14 x r = 29.5

r = 29.5 / (2 x 3.14)

r = 4.69

Surface area of basket ball = 4Πr²

= 4 x 3.14 x (4.69)²

= 276.27  cm²

So, the surface area of the basket ball is 276.27 cm²

Problem 8 :

A bowl has the form of a hollow hemisphere of radius 8.4 cm. Find the external surface area and the volume of the bowl. Give your answers to 0 significant figures. Take π = 3.142

Solution :

Radius = 8.4 cm

External surface area = 4Πr²

= 4 x 3.14 x 8.4²

= 886.23 cm²

So, the external surface area of the sphere = 886.23 cm²

Volume of hemispherical bowl = (2/3)Πr³

= (2/3) x 3.14 x 8.4²

= 147.70 cm³

So, volume of the sphere = 147.70 cm³

Problem 9 :

Find the radius of an opened hemisphere whose surface area is 1762 cm² Give your answers to 3 significant figures. Take π = 3.142

Solution :

Surface area of the hemisphere = 1762 cm²

4Πr² = 1762 cm²

4 x 3.14 r² = 1762

 r² = 1762 / (4 x 3.14)

r = 13.67

So, the radius of the sphere = 13.67 cm

Problem 10 :

A sphere has radius 12.6 cm. Find the volume and surface area. Give your answers to 3 significant figures. Take π = 3.142

Solution :

Volume of the sphere = (2/3)Πr³

r = 12.6

= (2/3)Π(12.6)³

= 4187.45 cm³

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