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Question 1

The diagonal of a quadrilateral is 30m in length and the length of the perpendiculars to it from the opposite vertices are 6.8 m and 9.6m. Find the area of the quadrilateral.

Solution:
ABCD be the given quadrilateral BE ⊥ AC and DF ⊥ AC.

Let AC = 30m, BE = 6.8m, DF = 9.6m.

Area of quadrilateral ABCD                    
          


 

Question 2

Find the area of a rhombus the lengths of whose diagonals are 36cm and 22.5cm.

Solution:
The area of the rhombus

                                     

Question 3

Find the area of a rhombus in which each side is equal to 15cm and one of whose diagonals is 24cm.

 


Solution:

Let ABCD be the rhombus

AB = 15cm, BO = 12cm.

Since AOB is right angled at O.

By pythagoras them,          

                        

Therefore the two diagonals are 18 cm and 24 cm.

The area of rhombus

Question 4

Find the area of a trapezium whose parallel sides are 57 cm and 39 cm and the distance between them is 28 cm.

Solution:
The area of the trapezium

                                       

Question 5

The area of trapezium is 352 cm2. The distance between the parallel sides is 16cm. If one of the parallel sides is 25 cm. Find the other.

 


Solution:
Area of a trapezium = 352cm2


 

Question 6

The parallel sides of a trapezium are 25cm and 13cm. Its nonparallel sides are equal each being 10cm. Find the area of the trapezium.

Solution:

 
AB = 25cm, DC = 13 cm, BC = 10 cm.

AD =10cm

EB = AB - AE = AB - DC

    = 25 - 13 = 12cm

CE = AD = 10cm. AE = DC = 13cm

In Δ EBC, CE = BC = 10cm

CF ⊥ AB, F is the midpoint of EB.

                           

Area of trapezium = area of parallelogram AECD + area of Δ CEB

                        

Question 7

Find the volume, the total surface area and the lateral surface of a cuboid which is 8m long, 6m broad and 3.5m high.

Solution:
The volume of cuboid

                                 

The total surface area of the cuboid

                                                  

The lateral surface area of the cuboid =



Question 8

Find the volume of a cube whose total surface area is 486cm2.


Solution:
Total surface area =

Length of each edge = 9cm.

Volume of the cube = .

Question 9

How many bricks will be required for a wall which is 8m long 6m high and 22.5cm thick if each brick measures 25cm × 11.25 cm × 6cm?

 


Solution:
The volume of the wall

Volume of 1 brick

.
 

Question 10

An open rectangular cistern when measured outside is 1.35 m long 1.08m broad and 90cm deep and is made of iron which is 2.5cm thick. Find the capacity of the cistern and the volume of the iron used.


Solution:
The external dimensions of the cistern are

Length = 135 cm, breadth = 108cm, depth = 90 cm.

External volume =

The internal dimensions of the cistern are

Length = (135 - 5) cm = 130 cm, breadth = (108 - 5) cm = 103cm

Height = (90 - 2.5) cm = 87.5 cm

The capacity of the cistern = internal volume of the cistern

Volume of iron = external volume - internal volume




Solution:
The area of the field               

The area of the pit

The area over which the earth is spread out         

The volume of earth dug out

∴ The rise in level .

Question 13

How many cubic metres of earth must be dug out to since a well is 16m deep and which has a radius of 3.5m? If the earth taken out is spread over a rectangular plot of dimensions what is the height of the platform so formed?


Solution:
The volume of the earth dug out

The area of the given plot

The volume of the platform formed = the volume of the earth dug out

                                                

The height of the platform

The height of the platform = 1.54m.

Question 14

An iron pipe is 21cm long and its exterior diameter is 8cm. If the thickness of the pipe is 1cm and iron weighs 8g/cm3. Find the weight of the pipe.


Solution:
The external radius of the pipe = 4cm

The internal radius of the pipe = (4 - 1) cm = 3cm

The external volume                 

The internal volume

The volume of the metal = external volume - internal volume

                                  

The weight of the pipe =

                               


Question 15

A closed metallic cylindrical box is 1.25m high and it has a base whose radius is 35cm. If the sheet of metal costs Rs. 80 per m2. Find the cost of the material used in the box. Find the capacity of box in litres.

Solution:

The area of metal used = total surface area of the box

     

                                

The cost of material used =

The capacity of the box = volume of the box

                                

Question 16

A rectangular piece of paper of dimensions 22cm by 12cm. is rolled along its length to form a cylinder. Find the volume of the cylinder so formed.


Solution:
Length of paper = Height of the cylinder = 12cm

Circumference of its base = 22 cm

                              

Volume of the cylinder

                               

Question 17

PQRS is a quadrilateral in which PQ =4 cms, QR = 9.2 cms, RS = 8cm SP = 6cm ∠ PSR = ∠ PQR = 90° . Find its area.


Solution:
Area of right angled

                                      

Area of right angled

                                   

Area of quadrilateral

                                     =18.4 + 24 = 42.4cm2

Question 18

Find the surface area of a chalk box whose length, breadth and height are 16cm, 8cm and 6cm respectively.


Solution:
Chalk box is in the form of cuboid.

Surface area of the cuboid

                                    

∴ The area of the chalk box is 544 cm2.

Question 19

The dimensions of a cuboid are in the ratio of 1:2:3 and its total surface area is 88m2. Find its dimensions.


Solution:
The dimensions of the cuboid are in the ratio 1:2:3

Let the dimensions be in metres

Surface area

∴ The dimensions are 2 m, 4 m and 6 m.

Question 20

The length of a hall is 20m and width 16m. The sum of the areas of the floor and the flat roof is equal to the sum of the areas of the four walls. Find the height and volume of the hall?

 


Solution:
Let 'h' be the height of the wall.

Sum of areas of four walls

Sum of the areas of the floor and the flat roof =

Given that the sum of the areas of four walls is equal to the sum of the areas of the floor and roof

Volume of the hall .





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