**Example 1 :**

A play top is in the form of a hemisphere surmounted on a cone.
The diameter of the hemisphere is 3.6 cm. The total height of the play
top is 4.2 cm. Find its total surface area.

**Solution :**

Let r and h are radius and height of the cone.

diameter of the hemisphere = 3.6 cm

radius of hemisphere (r) = 3.6/2 = 1.8 cm

total height of the play top = 4.2cm

radius of hemisphere + height of cone = 4.2

1.8 + h = 4.2

h = 4.2-1.8

h = 2.4 cm

Now we are going to find slant height of cone

l^{2} = r^{2}+h^{2}

= (1.8)^{2}+ (2.4)^{2}

= 3.24 + 5.76

l^{2} = 9

L = √(3 ⋅ 3)

L = 3 cm

To find the total surface area of the play top, we have to find the sum of curved surface area of hemisphere and curved surface area of cone.

Total surface area of play top = C.S.A of hemisphere + C.S.A of cone

= 2Πr^{2}+Πrl

= Πr(2r+l)

= Π ⋅ (1.8) [2(1.8) + 3]

= Π ⋅ (1.8) (3.6+3)

= Π (1.8) (6.6)

= 11.88 Π cm^{2}

Total surface area of the play top = 11.88 Π cm².

**Example 2 :**

A solid is in the shape of a cylinder surmounted on a hemisphere. If the diameter and the total height of the solid are 21 cm,25.5 cm respectively, then find its volume.

**Solution :**

Let r and h are radius and height of the cylinder.

diameter of the hemisphere = 21 cm

radius of hemisphere (r) = (21)/2 = 10.5 cm

Total height of the solid shape = 25.5 cm

radius of hemisphere + height of the cylinder = 25.5

10.5 + h = 25.5

h = 25.5-10.5

h = 15 cm

To find the volume of the solid shape we have to find the sum of volume of hemisphere and volume of cylinder.

Volume of solid toy = Volume of hemisphere + Volume of cylinder

= (2/3) Πr^{3} + Π r^{2}h

= Πr^{2}[(2/3) r + h]

= (22/7) x (10.5)² [(2/3) 10.5 + 15]

= (22/7) ⋅ 10.5 ⋅ 10.5 (2(3.5)+15)

= 22 ⋅ 1.5 ⋅ 10.5 ⋅ (7+15)

= 7623 cm^{3}

Volume of solid shape = 7623 cm^{3}.

**Example 3 :**

A capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. If the length of the entire capsule is 14 mm and diameter of the capsule is 5 mm, find its surface area.

**Solution :**

diameter of the hemisphere = 5 mm

radius of hemisphere (r) = 5/2 = 2.5 mm

total height of the solid = 14 mm

2(radius of hemisphere)+height of cylinder = 14

2 (5/2)+h = 14

5+h = 14

h = 14-5

h = 9 mm

Now we have to find total surface area of capsule. For that we have to find the sum of surface areas of two hemispheres and one cylinder.

Total surface area of capsule

= 2(CSA of hemisphere) + CSA of cylinder

= 2(2 Π r^{2})+2Πrh

= 2Πr(2r+h)

= 2⋅ (22/7)⋅ (2.5) [2(2.5)+9]

= (22/7) ⋅ 5⋅ (5+9)

= (110/7) (14)

= 220 mm^{2}

Total surface area of capsule = 220 mm^{2}.

**Example 4 :**

A tent is in the shape of right circular cylinder surmounted by a cone. The total height and diameter of the base are 13.5 m and 28 m. If the height of the cylindrical portion is 3 m, find the total surface area of the tent.

**Solution :**

The total height of the tent = 13.5 m

The diameter of the base of the tent = 28 m

radius of the tent = 14 m

Height of cylindrical portion = 3 m

Total height of the tent = 13.5 m

Height of cylinder + Height of cone = 13.5

3+h = 13.5

h = 13.5-3

h = 10.5 m

height of cylindrical portion is 10.5 m

l^{2} = r^{2}+h^{2}

= 14^{2}+10.5^{2 }

= 196+110.25

l = √306.25

l = 17.5

Total surface area of tent = CSA of cylinder + CSA area of cone

= 2Πrh + Πrl

= Πr(2h+l)

= (22/7) ⋅ 14 [2(3)+17.5]

= 22 ⋅ 2(6+17.5)

= 44 (23.5)

= 1034 cm^{3}

**Example 5 :**

Using clay, a student made a right circular cone of height 48 cm and base radius 12 cm. Another student reshapes it in the form of a sphere. Find the radius of the sphere.

**Solution :**

Height of the right circular cone (h) = 48 cm

Radius of right circular cone (r) = 12 cm

Volume of right circular cone = Volume of sphere

(1/3) Πr^{2}h = (4/3) Πr^{3}

(1/3) r^{2}h = (4/3) r^{3}

(1/3) (12)^{2}(48) = (4/3) r^{3}

(1/3) ⋅ 12 ⋅ 12 ⋅ 48 = (4/3) r^{3}

r^{3} = (1/3) ⋅ 12 ⋅ 12 ⋅ 48 ⋅ (3/4)

r^{3} = 3 ⋅ 12 ⋅ 48

r = ∛ (3⋅3⋅4⋅4⋅4⋅3)

r = 12 cm

Radius of sphere is 12 cm.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

**v4formath@gmail.com**

We always appreciate your feedback.

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**