Measuring Regular and composite solids (Class- 9, Experience- 8) - Active Math Class

NCTB Math Solution (English Version)

Measuring Regular and composite solids (Class- 9, Experience- 8)



[Exercise- Page 208]

E.8.1  A 12 cm long rectangular carrot tip has a diameter of 2.5 cm. To eat the carrot, you peel the carrot. What is the area of all the shells? Determine the nutritional content of the carrot.



Solution:


 12 cm long rectangular carrot tip has a diameter of 2.5 cm. Find the volume of the carrot-

Given,
The height of the carrot, h = 12 cm
The diameter of the ground towards the tip of the carrot is 2.5 cm,
Hence, the radius, r =  2.5   2 = 1.5 cm
The carrot is cone shape


We Know,
Volume of cone =  1  3 πr2h cubic unit

Volume of cone shape carrot =  1  3 πr2h cubic cm    [Here, π = 3.1416]

= { 1  3 ✕ 3.1416 ✕ (1.5)2 ✕ 12} cubic cm

= (3.1416 ✕ 1.5625 ✕ 4) cubic cm

= 19.635 cubic cm.


So, the volume of the carrot is 19.635 cubic cm (Answer)





E.8.2  As shown in the picture, the total surface area of a cuboid made of plastic found on the road is 1256.64 square cm, and the slant height is 26 cm. 


i) If it costs 1.50 taka per square centimeter to paint the curved surface of the cuboid with red color, how much will it cost in total? 

ii) How much plastic is there in the cuboid?

Solution:


(i) Find the total cost of painting on the curved surface of the cuboid-

It is seen that the cuboid used in the road is cone shape.

Given, In the cone shape cuboid, 

Base area, πr2 = 1256.64 square cm [We Know, the base area of cone = πr2 square unit, Here, r = base radius]
Slant height, l = 26 cm

Now,
πr2 = 1256.64

or, r2 =  1256.64      π

or, r2 =  1256.64   3.1416  [Here, π = 3.1416]

or, r2 = 400

or, r =  400 

or, r = 20 cm.


We know,
Area of curved surface of cone = πrl square unit

Since the cuboid is a cone shape, area of curved surface of the solid,
= πrl

= (3.1416 ✕ 20 ✕ 26) square cm  [Given, l = 26 cm]

= 1633.632 square cm.


To paint the curved surface of the solid,
1 square centimeter cost 1.50 taka
1633.632 sq cm costs (1.50 ✕ 1633.632) taka = 2450.448 taka.


So, the cost of painting the curved surface of the cube 2450.45 taka. (Answer)




(ii) Determine how much plastic is in the cuboid-

It is seen that cuboid made of plastic used on roads is cone shape.

Given,
in the cone shape cuboid, 
radius of the base, r = 20 cm  [from solution no.1]
Slant height, l = 26 cm

How much plastic is in the cuboid can be known by determining its volume.

According to the Pythagorean formula,
l2 = h2  + r2   [Here, l = slant height, h = height, r = base radius]

or, h2 = l2 - r2

or, h2 = 262 - 202

or, h2 = 676 - 400

or, h2 = 276

or, h =  276 

or, h = 16.6133 cm.



We Know,
Volume of the cone =  1  3 πr2h cubic unit

Since the plastic cuboid is cone shape, the volume of the cuboid,
 1  3 πr2h

= ( 1  3 ✕ 3.1416 ✕ 202 ✕ 16.6133) cubic cm

= ( 1  3 ✕ 3.1416 ✕ 400 ✕ 16.6133) cubic cm

= 6958.957 cubic cm (approx.)


So, the volume of plastic in the cuboid is 6958.957 cubic cm (approx.)





E.8.3  A plastic solid sphere has a radius of 6 cm. If the sphere is melted into a hollow sphere of radius 7 cm, find the plastic density of the hollow sphere.

Solution:


 If a solid sphere of radius 6 cm is melted into a hollow sphere of radius 7 cm, find the plastic density of the hollow sphere-

We Know,
volume of sphere =  4  3 πr3 cubic unit; Here, r = radius of sphere.

Now, volume of sphere (with radius 6 cm)
 4  3 ✕ 3.1416 ✕ 63 cubic cm


= 1.3333 ✕ 3.1416 ✕ 216  

= 904.7808 cubic cm.


and, volume of sphere (with radius 7 cm)
 4  3 ✕ 3.1416 ✕ 73 cubic cm
= 1.3333 ✕ 3.1416 ✕ 343  

= 1436.7584 cubic cm.



Now, a sphere of radius 7 cm is made from a sphere of radius 6 cm, so the sphere of radius 7 cm is hollow-
Volume of hollow part of sphere of radius 7 cm
= 1436.7584 cubic cm – 904.7808 cubic cm
= 531.9776 cubic cm.



Now find the radius r of the hollow part from the volume of the hollow part of a sphere of radius 7 cm,
 
      4  3 πr3 = 531.9776

or,  4  3 ✕ 3.1416 ✕ r3 = 531.9776

or, r3 = 531.9777 ✕          3  4 ✕ 3.1416  

or, r3 =     1595.9331   12.5664

or, r3 = 127

or, r =  127 

or, r = 5.0265 cm (approx.)



thickness of a sphere of radius 7 cm

= (7 - 5.02652) cm 

= 1.97348 cm (approx.)



So, the plastic density of the hollow sphere is 1.97348 cm (approx.) (Answer)





E.8.4  The radii of four perfect spheres are 3 cm, 8 cm, 13 cm and r cm respectively. What is the value of r if four spheres are melted to form a new solid sphere of radius 14 cm?

Solution:


The radii of four perfect spheres are 3 cm, 8 cm, 13 cm and r cm. Find the value of r if four spheres are melted to form a new solid sphere of radius 14 cm-


Given,
four solid spheres have radii of 3 cm, 8 cm, 13 cm ও r cm.
Four spheres are melted to form a new perfect sphere of radius 14 cm.


We Know, volume of sphere =  4  3 πr3 cubic unit [radius r]

Now, by condition,
volume of four fixed spheres = volume of a new sphere formed by four spheres

or, 43π33 + 43π83  + 43π133 + 43πr3 =  4  3 π143

or,  4  3 π(33 + 83  + 133 + r3) =  4  3 π143

or, (33  + 83  + 133 + r3) = 143  [both sides divided by  4  3π]

or, 27 + 512 + 2197 + r3 = 2744

or, 2736 + r3 = 2744

or, r3 = 2744 – 2736

or, r3 = 8

or, r =  8 

∴ r = 2


So, the value of r is 2 cm (Answer)




E.8.5  An equilateral heptagonal prism shaped aquarium has each side of length 25 cm and height 1 m. If it costs 2 taka per square centimeter to cover the lateral surface of the aquarium with glass, how much will it cost in total? How many liters of water will be required to fill three fourths of the aquarium, given that 1000 cubic centimeters = 1 liter?

Solution:


Calculate the total cost of covering the sides of the aquarium with glass. And how many liters of water will be required to fill three-fourths of the aquarium-
And how many liters of water will be required to fill three-fourths of the aquarium-

Given, in aquarium, 
the length of each side, a = 25 cm
height, h = 1 meter = 100 cm
equilateral heptagonal prism shape, n = 7


We Know,
Total surface area of the prism = 2 ✕ (area of the base) + area of all sides

The aquarium (prism) Area of all sides 
= (na ✕ h) square cm  
= 7 ✕ 25 ✕ 100 square cm  [put values]
= 17500 square cm.


Now,
To cover side surface of the aquarium with glass
1 sq cm costs 2 taka
17500 square cm costs (2 ✕ 17500) taka = 35000 taka.


 Again, We Know,
Volume of prism = surface area ✕ height

and, 
Area of regular polygon = (na2 4) cot(180°  n)   [n number of arms of length a]


Volume of aquarium = surface area ✕ height

= ( na2  4) cot(180°  n) ✕ h

= (7 ✕ 252     4) cot(180°  7) ✕ 100

= 441 ✕ cot(25.7143) ✕ 100

= 91574.5936  cubic cm.



Now, the volume of the aquarium is one third
 1  3 ✕ 91574.5936  cubic cm.
= 30521.812 cubic cm.


Now,
To fill 1000 cubic cm water required  1 liter
to fill 1 cubic cm water required     11000 liter
∴ 30521.812  cubic cm to fill water required  (   11000 ✕ 30521.812) liter = 30.5218 liter.

So, it will cost a total of Rs 35000 to cover the sides of the aquarium with glass and 30.5218 liters of water will be required to fill three-fourths of the aquarium. (Answer)





E.8.6  Each side of the plane of the equilateral prism in the figure is 5 cm and the sides are square.


i) Measure the surface area of the prism. 

ii) What is the curved surface area of the prism? 

iii) Determine the volume of the prism.

Solution:


(i) Find the surface area of the prism-

From the figure,
the number of sides of the prism, n = 8
length of each side of the surface, a = 5 cm

We Know,
The surface area of the prism = ( na2  4) cot(180°  n)

= ( na2  4 ) cot(180°  n)

= ( 8 ✕ 52     4) cot(180°  8)

= ( 8 ✕ 25     4) cot(180°  8)

= (2 ✕ 25) ✕ cot(22.5°)

= 50 ✕ 2.4142

= 120.710678 square cm (approx.)



Surface area of the prism

= (2 ✕ 120.7106) square cm (approx.)

= 241.4213 square cm (approx.)


So, Surface area of the prism 241.4213 square cm (approx.). (Answer)




(ii) Find the curved surface area of the prism-

Given,
the sides of the prism are square, therefore, the length of each side of the side, a = 5 cm.
And the height of the prism, h = 5 cm.

Area of curvature of the prism = nah square cm
= nah square cm

= (8 ✕ 5 ✕ 5) square cm

= 200 square cm

So, area of the curved surface of prism 200 square cm. (Answer)




(iii) Find the volume of the prism-

Volume of the prism = surface area ✕ height

= 120.710678 ✕ 5 cubic cm   [Substituting value from (i) no.]

= 603.55339 cubic cm (approx.)

So, volume of the prism 603.55339 cubic cm (approx.). (Answer)





E.8.7  A tent is constructed on a square ground of length 82 m by placing a pole of 66  high right in the middle. 

(i) Determine the length of the diagonal of the prism. 

(ii) How much money needs to be spent to purchase fabric at the rate of 200 taka per square meter?

(iii) Calculate thee volume of the empty space inside the prism?

Solution:


(i) Determine the length of the diagonal of the prism.


Given,
the length of the square base of the tent, a = 8 2  meter.
Height of pole at center, h =  66  meter.
Length of edge of tent = ?

The pole is located at the intersection of two diagonals of the square ground or at the midpoint of any diagonal.

We Know,
length of diagonal of square = a 2  [Here, length of side of square = a]
So, length of diagonal of square base = 8 2 . 2  meter = 16 meter.
and, half of length of diagonal =  16   2 meter = 8 meter.

Consider the following figure-

Here, in triangle ABC-
The hypotenuse of ABC is the length of the side of the tent-
The height of the middle pole, AB = h =  66  meter.
The length of the base of the triangle which is half of the hypotenuse, BC = 8 meter
The hypotenuse is the length of the side of the tent, AC


According to the Pythagorean formula,
AC2 = AB2 + BC2
or, AC2 = ( 66 )2 + (8)2
or, AC2 = 66 + 16
or, AC2 = 82
or, AC =  82 
or, AC = 9.0554 meter


So, the length of the edge of the tent is 9.0554 meter (Answer)




(ii) Determine how much money is needed to purchase fabric-

Consider the following figure-
Here, in triangle ADC-
inclined height, AD = S meter
side length of the tent is hypotenuse of the triangle, AC =  82  meter
The ground length of the triangle is half of the ground length of the tent, DC =  1  2 ✕ 8 2 

According to the Pythagorean theorem,
AC2 = AD2 + DC2

or, DC2 = AC2 - AD2

or, S2 = ( 82 )2 - { 1  2 (8 2 )}2 [put values]

or, S2 = 82 - ( 1  4 ✕ 64 ✕ 2)

or, S2 = 82 - 32

or, S2 = 50

or, S =  50 

or, S = 7.0711 meter



Now, the shape of the tent is similar to a pyramid

Lateral surface area of the pyramid

=  1  2(Ground Range ✕ inclined height) sq. unit

= ( 1 2  ✕ 4 ✕ 8 2  ✕ 7.0711) sq. meter  [put values]

= (2 ✕ 8 ✕ 1.4142 ✕ 7.0711) sq. meter

= 160 sq. meter.


Now,
Cloth for 1 sq. meter cost 200 taka
cloth for 160 sq. meter (200 ✕160) taka = 32000 taka 


So, clothes cost 32000 taka. (Answer)




(iii) Determine the amount of air space found in the tent.

We Know,
Volume of pyramid = ( 1  3 ✕ surface area ✕ height) cubic unit


the volume of the tent

 1  3 ✕ (8 2 )2 ✕  66  cubic meter [put values]

 1  3 ✕ 64 ✕ 2 ✕  66  cubic meter

 1  3 ✕ 64 ✕ 2 ✕ 8.1240 cubic meter

= 346.624 cubic meter (approx.)


So, approximately 346.624 cubic meters of airy void space was found within the tabernacle. (Answer)






E.8.8  A pyramid with a base of a square of side 6 meters and a slant height of 67 meters is situated on the square land. 

(i) Determine the height of the pyramid. 

(ii) What is the total surface area of the pyramid? 

(iii) Calculate the volume of the pyramid.

Solution:


(i) Find the height of the pyramid.

Given,
perimeter of the pyramid, = 67 meter
length of side of the ground = 6 meter

Since the ground of the pyramid is square, the position of the lower point of the altitude line will be found at the midpoint of the length of the diagonal of the ground.

We Know, length of diagonal of square = a 2  [Here, length of side of square is a]

Hence, length of hypotenuse of square plot = 6 ✕  2  = 6 2  meter.

Length of half hypotenuse of square = 12 ✕ 6 2  meter = 3 2  meter.


Now let's look at the image below,

 Here, ABC is a right triangle-
Side of pyramid or hypotenuse of triangle, AC = 67 meter
Length of half hypotenuse of square area or area of triangle, BC = 3 2  meter
Altitude of pyramid or altitude of triangle, AB = ?

Now, according to the Pythagorean theorem,
AC2 = AB2 + BC2
or, AB2 = AC2 - BC2
or, AB2 = 672 - (3 2 )2
or, AB2 = 4489 - 9 ✕ 2
or, AB2 = 4489 - 18
or, AB2 = 4471
or, AB2 =  4471 
or, AB = 66.8655 meter.


So, the height of the pyramid is 66.8655 meters.



(ii) Find the total surface area of the pyramid.

Consider the figure below,

Here, in right triangle ADC-
side of the pyramid or hypotenuse of the triangle, AC = 67 meter
Half of the base length of the pyramid or base length of the triangle, BC =  6  2 meter = 3 meter
Let, slant height of pyramid or height of triangle, AD = S meter

According to Pythagoras formula,
AC2 = AD2 + DC2
or, AD2 = AC2 - DC2
or, S2 = 672 - 32
or, S2 = 4489 - 9
or, S =  4480 
or, S = 66.9328 meter


The slant height height of the pyramid, S = 66.9328 meter.


We Know,
the total surface area of the pyramid

= surface area +  1  2 (perimeter of base ✕ slant height) square unit

= 62  +   1  2 (4 ✕ 6 ✕ 66.9328) sq. meter

= (36  +  2 ✕ 6 ✕ 66.9328) sq. meter

= (36 + 803.1936) sq. meter

= 839.1936 sq. meter (approx.)



So, the total surface area of the pyramid is 839.1936 sq. meter (approx.). (Answer)




(iii) Find the volume of the pyramid.

From answer to question no. (i),
height of height = 66.8655 meter.
Side length of square = 6 meter


Volume of pyramid 
 1  3 (surface area ✕ height) cubic meter

= ( 1  3 ✕ 62 ✕ 66.8655) cubic meter

= ( 1  3 ✕ 36 ✕ 66.8655) cubic meter

= 802.386 cubic meter



So, the volume of the pyramid is 802.386 cubic meter. (Answer)





E.8.9  The given solid in the figure has a base with a radius of 4 meters in the lower part and a height of 5 meters. The sloping height in the upper part is 3 meters. 
(i) If it costs 450 taka per square meter to paint the curved surface of the solid in the lower part, how much will it cost in total? 

(ii) What is the total surface area of the solid? 

(iii) Determine the volume of the solid.

Solution:


(i) It will cost to paint the curved surface of the lower part of the cube-

The lower part of the cube is like a cylinder as shown in the figure
Given, diameter of cylinder = 4 meter
Radius of cylinder, r =  4  2 meter = 2 meter
and height of cylinder, h = 5 meter


We Know,
Area of curved surface of cylinder
= 2πrh sq. meter
= 2 ✕ 3.1416 ✕ 2 ✕ 5 sq. meter  [π = 3.1416]
= 62.832 sq. meter


Now, the cost of painting of curved surface
1 square meter is 450 taka
62.832 sq. meter is (450 ✕ 62.832) taka = 28274.4 taka.


So, the cost of painting the curved surface of the lower part of the solid will be 28274.4 taka. (Answer)





(ii) Find the total surface area of the solid-

The solid is formed by a cone and a cylinder where the circumference of the cylinder and the area of the cone are the same.
Given, the base radius of the cylinder or the base radius of the cone, r = 2 meter
the inclined height of the cone, l = 3 meter

Here,
Area of the solid
= Area of the curved surface of the cone + Area of the curved surface of the cylinder + Area of the base of the cylinder [since the base of the corner is connected between the cylinders no calculations needed]

= πrl + 62.832 + πr2  [value of area found from (i) no. answer]

= 3.1416 ✕ 2 ✕ 3 + 62.832 + 3.1416 ✕ 22   sq. meter

= 18.8496 + 62.832 + 12.5664 sq. meter

= 94.248 sq. meter


So, the total surface area of the cube is 94.248 sq. meter. (Answer)




(iii) Find the volume of the solid-

For cylinder,
base radius, r = 2 meter
height, h = 5 meter


For cone,
base radius r = 2 meter
inclined height l = 3 meter


If the height of the cone is h1 ,
l2 = h12  + r2

or, 32 = h12  + 22

or, 9 = h12  + 4

 or, h12 = 5

or, h1 =  5 



Volume of solid
= volume of cylinder + volume of cone

= πr2h +   1  3 πr2h1

= 3.1416 ✕ 22 ✕ 5 +  13 ✕ 3.1416 ✕ 22 5

= 62.832 + 9.36644

= 72.19844 cubic meter (approx.)



So, the volume of the solid is 72.19844 cubic meter. (Answer)






E.8.10  The solid given in the figure is placed on a rectangular base with dimensions of length 6 meters and width 4 meters. The height of the lower part is 7 meters. The length of the diagonal in the upper part is 7.5 meters. 
(i) If it costs 2250 taka per square meter to apply iron sheets on the lateral surface of the lower part of the solid, how much will it cost in total? 

(ii) Determine the lateral surface area of the upper part of the solid. 

(iii) Determine the volume of the solid.

Solution:


(i) Find the total amount required to put iron sheet around the lower part of the block-

Given, lower part of compound block is rectangular shape,
length of the rectangular block = 6 meter
width of the rectangular block = 4 meter and
height of the rectangular block = 7 meter.


Area of four sides of rectangular cube
= perimeter of land ✕ height
= 2(6 + 4) ✕ 7
= 20 ✕ 7
= 140 sq. meter


Now,
to put iron sheet around the bottom of the solid,
1 sq. meter cost 2250 taka
140 sq. meter cost (2250 ✕ 140) taka = 315000 taka.


So, the cost of iron sheeting around the lower part of the enclosure is 315000 taka. (Answer)





(ii) Find the area of the upper part of the cube-

Given, 
 the upper part of the compound cube is an inverse pyramid, consisting of two inverted triangles,
One with base length = 4 meter 
and the other with base length = 6 meter.
Length of each edge = 7.5 meter;

Hence, each of the triangles has two equal sides, i.e. the triangles are isosceles triangles.

We Know,
Area of an isosceles triangle =  b  4  (4a2 - b2 [where, a is an isosceles and b is an isosceles]


Here,
The area of two inverted triangles with bases of 6 meters of the pyramid

= 2 ✕  6  4  4 ✕ (7.5)2 - 62 

= 3 4 ✕ 56.25 - 36 

= 3 225 - 36 

= 3 189 

= 3 ✕ 13.7477

= 41.243 sq. meter (approx.)



Again,
The area of the pyramid with two inverted triangles of 4 m base

= 2 ✕  4  4  4 ✕ (7.5)2 - 42 

= 2 ✕  4 ✕ 7.5 - 16 

= 2 ✕  4 ✕ 56.25 - 16 

= 2 ✕  2 09

= 2 ✕ 14.4568

= 28.914 sq. meter (approx.)



Surface area of pyramid
= (6 ✕ 4) sq. meter
= 24 sq. meter


Area of top of cube
= (28.914 + 41.243 + 24) sq. meter
= 94.159 sq. meter (approx.)


So, the area of top of cube is 94.159. (Answer)




(iii) Find the volume of the cube-

Volume of the cube = Volume of the inhomogeneous pyramid + Volume of prism

Now, in determining the volume of the pyramid, if a perpendicular to the ground is drawn from the top of the equilateral pyramid, it will fall at the midpoint of the diagonal of the ground. Consider the following figure-

Length of the base diagonal of the pyramid =  62 + 42  meter =  36 + 16 =  52 =  4 ✕ 13 = 2 13 meter.


As per figure,
BC =  13  meter; 
AC = 7.5 meter, 
Height of pyramid, AB

According to Pythagoras formula,
AB2 = AC2 – BC2
or, AB2 = (7.5)2 – ( 13 )2
or, AB2 = 43.25
or, AB =  43.25 
or, AB = 6.5765 meter


Volume of inhomogeneous pyramid
 1  3 ✕ surface area ✕ height

= ( 1  3 ✕ 6 ✕ 4 ✕ 6.5765) cubic meter

= (3 ✕ 4 ✕ 6.5765) cubic meter

= 78.918 cubic meter (approx.)


and,
Volume of the prism
= surface area ✕ height
= (6 ✕ 4 ✕ 7) cubic meter
= 168 cubic meter


Volume of the compound cube = (78.918 + 168) cubic meter
                                              = 220.611 cubic meter.


So, the volume of the composite volume is 220.611 cubic meter. (Answer)





E.8.11  The solid given in the figure has a base radius of 10 centimeters and a height of 16 centimeters in the lower part. 

(i) What is the height of the geometric solid? 
(ii) Determine the surface area of the upper part of the geometric solid. 

(iii) Calculate the total surface area of the geometric solid. 

(iv) Determine the volume of the geometric solid.

Solution:


(i) Find the height of the geometric solid-

We can see that the upper part of the geometric solid is like hemisphere and lower part of geometric solid is like cylinder.

Given,
 Height of hemisphere = radius = 10 cm.
(Since the cube is a hemisphere, its radius will be the height of the hemisphere)
And the height of the lower part or cylinder shape = 16 cm.

Here,
height of cube = 10 + 16 cm = 26 cm.


So, the height of the geometric solid is 26 cm. (Answer)



(ii) Find the area of the upper part of the geometric solid-

Given,
the top of the solid is a hemisphere of radius, r = 10 cm

Need to find-
Area of the upper hemispherical part of the solid = Area of the surface of the hemisphere + Area of the base of the hemisphere

We Know,
Area of sphere Surface = 4πr2 square unit


Hence, Surface area of hemisphere
= 2πr2 square cm
= (2 ✕ 3.1416 ✕ 102) square cm
= (2 ✕ 3.1416 ✕ 100) square cm
= 628.32 square cm


Again,
Surface area of hemisphere
= πr2 square cm
= (3.1416 ✕ 102) square cm
= (3.1416 ✕ 100) square cm
= 314.16 square cm


Area of hemisphere = (628.32 + 314.16) square cm = 942.48 square cm.


So, the surface area of the upper part of geometric solid is 942.48 square cm. (Answer)




(iii) Find the total surface area of the geometric solid-

Given,
height of the bottom of the solid or cylinder part = 16 cm.
Radius of the bottom of the cube or cylinder part = 10 cm
Height of the top of the cube or hemisphere part = radius = 10 cm.


Total surface area of a solid
= Surface area of hemisphere + Surface area of cylinder + Area of cylinder base

= 2πr2  + 2πrh + πr2 square cm

= (2 ✕ 3.1416 ✕ 102  + 2 ✕ 3.1416 ✕ 10 ✕ 16 + 3.1416 ✕ 102) square cm

= (2 ✕ 3.1416 ✕ 100  + 2 ✕ 3.1416 ✕ 10 ✕ 16 + 3.1416 ✕ 100) square cm

= (628.32 + 1005.312 + 314.16) square cm

= 1947.792 square cm


So, the total surface area of the geometric solid is 1947.792 square cm. (Answer)





(iv) Find the volume of the geometric solid-

Volume of the solid = Volume of the hemisphere + Volume of the cylinder

We Know,
volume of sphere =  4  3 πr3 cubic unit.
Volume of the hemisphere =  4  6 πr3 cubic unit.

and,
Volume of the cylinder = πr2h cubic unit.


Volume of the solid
= (46 πr3 + πr2h) cubic unit

= (46✕3.1416✕103 + 3.1416✕102✕16) cubic cm

= (23✕3.1416✕1000 + 3.1416✕100 ✕16) cubic cm

= 2094.3791 + 5026.56

= 7120.96 cubic cm


So, volume of the geometric solid is 7120.96 cubic cm. (Answer)





E.8.12  Notice the geometric solid shown in the figure carefully. 
(i) What is the length of the inclined base of the solid? 

(ii) Determine the surface area of the upper part of the geometric solid. 

(iii) Determine the total surface area of the geometric solid. 

(iv) Determine the volume of the geometric solid.

Solution:


(i) Find the length of the inclined base of the solid-

Given,
Upper part of the geometric solid is like cone shape, where,
height, h = 10 cm;
base radius, r = 10 cm
and its inclined height, l, then


According to Pythagoras formula,
l2 = h2 + r2
or, l2 = 102 + 102  
or, l2 = 200
or, l =  2 00
or, l = 14.1421 cm (Appox.)


So, the length of the inclined base of the solid is 14.1421 cm (Appox.). (Answer)



(ii) Find the surface area of the upper part of the geometric solid-

Given,
Upper part of the geometric solid is like cone shape, where
height, h = 10 cm;
base radius, r = 10 cm
and its inclined height, l = 14.1421 cm (Appox.) [value from solution no. (i)]


Total surface area of cone shaped upper part of the geometric solid

= (πr2  + πrl) square unit

= (3.1416 ✕ 102  + 3.1416 ✕ 10 ✕ 14.142) square cm

= (314.16 + 444.2851) square cm

= 758.4493 square cm (approx.)


So, the surface area of the upper part of the geometric solid is 758.4493 square cm (approx.)




(iii) Find total surface area of the geometric solid-

Given, upper part of the geometric solid is like cone shape and lower part of the geometric solid is like hemisphere shape, where,
radius of hemisphere and radius of cone, r = 10 cm
inclined height of the cone, l = 14.142 cm (Appox.)  [value from solution no. (i)]


Total surface area of geometric solid
= Area of the cone + Area of the hemisphere

= (πrl + 2πr2) square unit

= (3.1416✕10✕14.142 + 2✕3.1416✕102) square cm 

= (444.2851 + 628.32) square cm

= 1072.609 square cm (approx.)


So, total surface area of the geometric solid is 1072.609 square cm (approx.)


(iv) Find the volume of geometric solid-

Given,
radius of hemisphere the radius of cone, r = 10 cm
inclined height of cone, l = 14.142 cm (Appox.) [value found from solution no. (i)]

The solid has two types of shape, the lower part is hemispherical and the upper part is angular.


Volume of the geometric solid

= Volume of cone + Volume of sphere

= ( 1  3 πr2h +   2  3 πr3) cubic unit

= ( 1  3 ✕ 3.1416 ✕ 102 ✕ 10 +   2  3 ✕ 3.1416 ✕ 103) cubic cm (approx.)

= (0.3333 ✕ 3.1416 ✕ 100 ✕ 10 +  0.66666 ✕ 3.1416 ✕ 1000) cubic cm (approx.)

= (1047.1895 +  2094.4) cubic cm (approx.)

= 3141.5895 cubic cm.


So, the volume of the geometric solid 3141.5895 cubic cm (Answer)




E.8.13  Notice the geometric solid shown in the figure carefully. 
(i) Determine the surface area of the upper part of the geometric solid. 

(ii) Determine the height of the solid. 

(iii) Determine the total surface area of the geometric solid. 

(iv) Determine the volume of the geometric solid.

Solution:


(i) Find the surface area of the upper part of the geometric solid-

From the figure,
The solid is made up of three shapes, the bottom is a hemisphere, the middle is a cylinder and the upper is an cone. Here, radius of hemisphere, cylinder and cone are same, r = 5 unit
Radius of upper part cone, r = 5 unit
Height of upper part cone, h1 = 12 unit

If slant height of upper part cone is l,

Area of cone at upper part
= (πr2  + πrl) square unit


Think a right triangle in the middle of upper part,
According to Pythagoras formula,
l2 = h12  + r2
or, l2 = 122  + 52
or, l2 = h2  + r2
or, l2 = 169
or, l = 13


Area of cone shape upper part
= (πr2  + πrl) square unit
= (3.1416✕52  + 3.1416✕5✕13) square unit
= (3.1416 ✕ 25  + 204.204) square unit

= (78.54 + 204.204) square unit
= 282.744 square unit

So, the area of the upper part of the geometric solid 282.744 square unit (Answer)



(ii) Find the height of the solid-

From the figure,
The solid is made up of three shapes, the bottom is a hemisphere, the middle is a cylinder and the upper is an cone. Here, radius of hemisphere, cylinder and cone are same, r = 5 unit
Height of upper part cone, h1 = 12 unit
Height of middle part cylinder, h2 = 5 unit
Height of lower part hemisphere, h3 = 5 unit [In sphere, the distance on all sides is equal to its radius]]


Consider the figure of geometric solid,
the sum of the heights of these three shapes is the height of the solid.

Height of geometric solid = (12 + 5 + 5) unit = 22 unit.


So, Height of geometric solid is 22 unit. (Answer)



(iii) Find the total surface area of geometric solid-

From the figure,
The solid has three surfaces, (i) surface area of cone, (ii) surface area of cylinder and (iii) surface area of hemisphare.
Radius of each, r = 5 unit
height of middle cylinder, h2 = 5 unit
slant height of upper cone, l = 13 unit [value obtained from solution (i) No]


Total surface area of solid
= Surface area of cone + Surface area of cylinder + Surface area of hemisphere

= πrl + 2πrh + 2πr2 square unit [We know, according to formula]

= (3.1416✕5✕13 + 2✕3.1416✕5✕5 + 2✕3.1416✕52) square unit

= (204.204 + 157.08 + 157.08) square unit

= 518.364 square unit


So, the total surface area of geometric solid is 518.364 square unit. (Answer)


(iv) Find the volume of geometric solid-

The solid is made up of three shapes, the bottom is a hemisphere, the middle is a cylinder and the upper is an cone. Here, radius of hemisphere, cylinder and cone are same, r = 5 unit
Height of upper part cone, h1 = 12 unit

Slant height of upper part cone, l = 13 unit [value obtained from solution no.(i)]
Height of middle part cylinder, h2 = 5 unit


Volume of geometric solid
= Volume of cone + Volume of cylinder + Volume of hemisphere

= ( 1  3 πr2l  + πr2h2 +   2  3 πr3) cubic unit [We know, according to formula]

= ( 1  3 ✕ 3.1416 ✕ 52 ✕ 13 + 3.1416 ✕ 52 ✕ 5 +   2  3 ✕ 3.1416 ✕ 53) cubic unit 

= (0.3333 ✕ 3.1416 ✕ 25 ✕ 13 + 3.1416 ✕ 25 ✕ 5 +  0.66666 ✕ 3.1416 ✕ 125) cubic unit

= 340.30596 + 392.7 + 261.79738) cubic unit

= 994.8033 cubic unit



So, the volume of geometric solid is 994.8033 cubic unit. (Answer)





E.8.14  In the figure, a hemisphere and a cone are placed accurately inside a cylinder. 

(i) Calculate the lateral surface area of the cone. 

(ii) Find the area of the curved surface of the hemisphere. 

(iii) Determine the volume of the hollow part inside the cylinder. 

(iv) What is the ratio of the volumes of the hemisphere, the cone, and the cylinder.

Solution:


(i) Find lateral surface area of the cone-

From the figure,
Height of cone, h = 12 cm and
Radius of base of cone, r = 6 cm.

Here, if the inclined height of the cone is l,

According to the Pythagorean formula,
l2 = h2 + r2
or, l2 = 122 + 62
or, l2 = 180
or, l =  180 
or, l = 13.4164


We know,
Area of curved surface of cone
= πrl square unit
= 3.1416 ✕ 6 ✕ 13.4164 square cm
= 252.8939 square cm (approx.)


So, the lateral surface area of cone is 252.8939 square cm (approx.). (Answer)




(ii) Find the area of the curved surface of the hemisphere-

From figure,
Radius of hemisphere, r = 6 cm

We Know,
Area of curved surface of hemisphere
= 2πr2 square unit
= (2 ✕ 3.1416 ✕ 62) square cm
= (2 ✕ 3.1416 ✕ 36) square cm
= 226.1952 square cm.


So, the area of the curved surface of the hemisphere is 226.1952 square cm (approx.). (Answer)




(iii) Find the volume of the hollow part inside the cylinder-

From figure,
The radius of the cylinder, the base of the cone and the radius of the hemisphere are same, r = 6 cm
Height of the cylinder, hc = (12 + 6) cm = 18 cm
Height of the cone, hk = 12 cm
Inclined height of the cone, l = 13.4164 cm


Volume of the hollow part of the cylinder = Volume of the cylinder - (Volume of the cone + Volume of the hemisphere)


Volume of the cylinder
= πr2h cubic unit [by formula]
= 3.1416 ✕ 62 ✕ 18 cubic cm
= 3.1416 ✕ 36 ✕ 18 cubic cm
= 2035.7568 cubic cm


Volume of the cone
 1  3 πr2h cubic unit [by formula]

 1  3 ✕ 3.1416 ✕ 62 ✕ 12 cubic cm

= 0.33333 ✕ 3.1416 ✕ 36 ✕ 12 cubic cm

= 452.3904 cubic cm



Volume of hemisphere
 2  3 πr3 cubic unit [by formula]

 2  3 ✕ 3.1416 ✕ 63 cubic cm

= 0.6666 ✕ 3.1416 ✕ 216 cubic cm

= 452.3904 cubic cm



Volume of hollow part of cylinder
= Volume of cylinder - (Volume of cone + Volume of hemisphere)
= 2035.7568 - (452.3904 + 452.3904) cubic cm
= 2035.7568 - 904.7808 cubic cm
= 1130.976 cubic cm


So, Volume of the hollow part inside the cylinder is 1130.976 cubic cm (approx.). (Answer)




(iv) Find the ratio of volumes of the hemisphere, the cone and the cylinder-

Here, from the solution no. (iii), we obtained,
Volume of hemisphere = 452.3904 cubic cm
Volume of cone = 452.3904 cubic cm
Volume of cylinder =2035.7568 cubic cm


Ratio of volume-
Volume of hemisphere : Volume of cone : Volume of cylinder
= 452.3904 : 452.3904 : 2035.7568
= 1 : 1 : 4.5 [divide each by 452.3904]
= 2 : 2 : 9 [multiply each by 2]

So, the ratio of volumes of the hemisphere, the cone and the cylinder is 2 : 2 : 9 (Answer)




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