Thursday, November 11, 2010

MENSURATION

Question 1
The area of a circular plate is one-sixteen the surface area of a ball. If the area of the plate is given as pcm2, then the radius of the ball is (UME 1981, Question 35)

Solution

Area of circular plate = pcm2
Let the area of ball be
Area of circular plate =  qcm2
Area of ball
               
Question 2






In the above diagram XZ is the diameter of a circle .  If XY is 4cm then the area of the triangle XYZ is  (UME 1981, Question 36)


 

Solution                                                                                                                                                                                                                                                                      





Question 3

A square of cardboard is taped at the perimeter by a piece of ribbon 20 cm long. What is the area of the board (UME 1979, Question 16)

Solution



 




Perimeter of a square = 4l
                                    4l =20cm
                                    l = 5cm
Area of square = l2 = 52 cm2 =25cm2

Question 4

A steel ball of radius 1cm is dropped into a cylinder of radius 2cm and height 4cm. If the cylinder is now filled water. What is volume in the cylinder?
(UME 1979, Question 20)

Solution

Volume of the sphere =
Given radius of the sphere is 1 cm
Volume of the sphere will be =
Volume of the cylinder =
Given radius of the cylinder as 2cm and height as 4cm.
Volume of the cylinder will be
The sphere will displace  water from the cylinder (Archimedes ‘s Principle)
The volume of water left will be

Question 5

A cylinder of height h and radius r is open at one end. Its surface area is
         

Question 6
In the parallelogram PQRS, PE is perpendicular to QR.  Find area of parallelogram




5
 
 











 Solution
               

Question 7
A Solid cylinder of radius 3cm has a total surface area of 36p cm2. Find its height (UME 1980, Question9)

Solution







Total surface area of cylinder =
                       
The height is 3cm

 

Question 8

In the below , , PS =7cm,  PT =7cm, SR = 4cm. Find the ratio of QRST to the area of PQRS (UME1980, Question 11)













 

Solution












Area of QRST = b ´ h = 4 ´7cm2 =28cm2
Area of PQRS = ½ ( 4 + 7)7 =77/2 cm2 
Ratio QRST : PQRS
            28: 77/2
            56: 77

Question 9

Find the area of curved surface of a cone whose base radius is 6cm and whose height is 8cm Take p= 22/7(UME 1980, Question 36)

Solution



 






              
              
Question 10
A cylinder motor of height 12cm has uniform thickness of 2cm. If the diameter of its outer cross –section is 10 cm. Find the volume of material (take p= 22/7)  (UME 1980, Question 44)

Solution



 










Volume of cylinder =pr2h

Volume of the inner cylinder  =

Volume of the outer cylinder =
Volume of the composite cylinder =
Volume of the composite cylinder =

Question 11

The area of the curved surface of the cone generated by the circle radius 6cm and arc length 22cm is 
Take p =22/ 7 (UME 1982, Question 39)

Solution









 








Radius of sector  = l = 6cm
The length of arc AXB is the same as the circumference of the circular base of the cone
                                         

Question 12

The square base of a pyramid of side 3cm has height 8cm. If the pyramid is cut into two parts by a plane parallel to the base midway between the base and vertex.. The volumes of the two sections are (UME 1982, Question 40)

Solution
 







Total volume of square based pyramid
=
Using similar triangle
Volume of pyramid AHBCD  =
The volume of the base section = (24 – 3) cm3
The volume of the base section = 21cm3
The volume of the two section are 21cm3, 3cm3

Question 13







PQRS is a parallelogram with area 50 square cm and side PQ is 10 cm long. T is appoint on RS and TF is the attitude of the triangle, Find TF (UME 1982 Question 44)

Solution

Area of Parallelogram = base ´ height
50cm2 = 10h
h = 5cm
where h = TF, TF =5cm

Question  14






What is the volume of the regular three-dimensional figure above? (UME 1984, Question 25)

Solution






Question 15
A cone is formed, by bending a sector having an angle of 120o. Find the radius of the base of the cone. If the diameter of the circle is 12cm

Solution









Radius of the sector = l = d/2 =12/2 cm = 6cm
The length of arc AXB is the same as the circumference of the circular cone
           

Question 16

Find the area of regular hexagon inscribed in a circle of radius 8cm (UME 1985, Question 36)

 

 


 

 

Solution



 












Since the polygon is a regular polygon, all the sides AB = AO =DB = 8cm

Question17
           
  (UME  1985, Question 42)

Solution








Area of trapezium =
                     

Question 18

How many  diameter disc can be cut out of sheet of cardboard  long and  wide (UME 1985 Question 45)

Solution

Area of the cardboard                             
            =
Area of disc =
The number of disc that could be from the cardboard will be


The figure is a solid with the trapezium PQRS as its uniform cross –section. Find its volume
 












Solution

Volume of block =Area of PQRS ´ 12cm3
Area of  PQRS =







With the construction of figure above
                       
Area of PQRS =
Volume of the block is

Question 19

Find the total surface area  of a solid cone of radius .(UME 1986, Question 41)

Solution

 







Question 20
An open rectangular box externally measures 4m ´ 3m ´ 4m. Find the total cost of painting the box       externally. If it cost N2.00 to paint one square metre (UME 1986, Question 44)
 

Solution











            Total area of the rectangular box =
Total cost of painting 68cm2 will be 68 ´ N2.00 = N 136.00

Question 21










In the figure above PQRS is rectangle, if the shaded area is 72sq.cm. Find h

Solution











Area of QAEP = 2´3h cm2 =6hcm2
Area of BRSF = 2 ´ 3h = 6h cm2
Area of ABCD =(2h – 4) ´ 2cm2 =(4h –8)cm2
Total Area of the shaded portion =(6h + 6h +4h – 8 )cm2 =72cm2
16h = 80,        Þ h =5 cm

Question 22

The base of a pyramid is a square of side 8cm. If its vertex is directly above the centre, find the height, given hat the edge is  (UME 1987,Question 43)

 

 

Solution


 







Using Pythagoras theorem
 
Question
In the figure, PS =7cm and RY = 9cm. If the area of parallelogram PQRS is 56cm. Find the area of trapezium. (UME 1988, Question 33)


 






















Area of parallelogram = b ´ h
Question 23
A quadrant of a circle of radius 6cm is cut away from each corner of a rectangle 25cm long and 18cm wide. Find the perimeter of the remaining figure. (UME 1988 Question 34)

 

Solution



 









Perimeter of the rectangle = 2(18 +25) cm = 86cm
Perimeter of the one quadrant =
        
Perimeter of the four quadrants =
                
Perimeter of the remaining figure is
            = (86 – 12p)cm

Question 24

If a metal pipe 10cm has an external diameter of 12cm and thickness of 1cm. Find the volume of the metal used in making the pipe (UME 1988, Question 44)

Solution



 




Volume of the cylinder =
Volume of internal cylinder=
Volume of external cylinder =
Volume of the pipe =

Question 25








In the figure above, a solid consist of hemisphere by a right circular cone with radius 3cm and height       6.0cm.Find the volume of the solid (UME 1988 Question 45)

Solution

Volume of cone =
Volume of hemisphere =
Volume of the solid shape

Question 26

A square tile has side 30cm. How many of these tiles will cover a rectangular floor of length 7.2m  and width 4.2m?  (UME 1989, Question 43)

Solution


Question  27
A cylindrical metal pipe 1m long has an outer diameter of 7.2cm and an inner diameter of 2.8cm. Find the volume of the metal used of the cylinder.(UME 1989, Question 44)

Solution



 




Volume of the metal =
R =3.6cm, r =1.4cm, h =100cm
Volume =

Question 28



 








OXYZW is a pyramid with a square base such that OX = OY = OZ = OW = 5m and XY =XW =YZ = WZ        = 6cm.     Find the height of OT (UME 1989 Question 45)

Solution









Question 29

The perimeter of rectangular lawn is 24cm, if the area of the lawn is 35m2.  How wide is the lawn


Solution

Perimeter = 2(l + w) =24
                        l+ w =12 – – – –(1)
Area of the rectangular lawn A =lw = 35m2 – – – (2)
From (1) l =12 – w
Substitute (12 – w) for l in equation (2)
(12 – w)w = 35
w2 – 12w + 35 = 0
(w – 7)(w – 5) =0
w = 7 or w = 5
The width should normally be less to the length of the lawn. So width will be 5m


 

Question 30





In the diagram, the area PQRS is 73.5cm2 and its height is 10.5cm. Find the length   of PS. If QR is one–third of PS (UME 1990, Question 31) 


 

Solution





Question 31














 

Solution










Using Pythagoras theorem

Question 32

A cylinder pipe made of metal is 3cm thick. If the internal radius is 10cm. Find the volume of the metal used in making 3m of the pipe (UME 1990, Question 42)

 

Solution



 





Radius of internal pipe = r =10cm
Radius of external pipe = R =13cm
Volume of the pipe =

Question

If the height of two circular are in the ratio 2:3 and their bases radii are in the ratio 9:8. What is the ratio of their volume (UME 1990, Question 43)

Solution

         

 

 

 

 

 

 

Question 33










Find the curve surface area of the frustum in the figure above (UME 1990 Question 44)

Solution














Using similar triangles
           


4cm
 
Q
 

U

 
Question 35









In the figure above PQRS is a square of sides’ 8cm. What is the area of  (UME 1991, Question 44)

 

Solution










Using Pythagoras

Question 36

Find the total area of the surface of a solid cylinder whose base radius is 4cm and height is 5cm (UME 1991, Question 45)

Solution

Question














Find the volume of the figure above (UME 1991, Question 46)

Solution

Question 38










The above figure is a circle with center O. Find the area of  the shaded portion (UME 1992 , Question30)
Area of Triangle =
Area of a quadrant =
Area of the shaped portion =

Question 39











In the figure above, the area of the square PQRS is 100cm2. If the ratio of the area of the square TUYS to the area of the square XQVU is 1:16, Find YR (UME 1993 Question 35)

Solution













Question 40
Find the radius of a sphere whose surface area is 154cm2 (  (UME 1993 Question 36)

Solution

Question 41

An open rectangular box is made of wood of 2cm thick. If the internal dimension of the box are 50cm long, 36cm wide and 20 cm deep. The volume of the box is (UME 1994, Question 29)

Solution



 






Internal dimensions are 50cm, 36cm and 20cn
External dimension are 54cm, 40cm and 21cm

 

 

 

 

 

Question 43



 






In the frustum of a cone shown above, the top diameter. If the height of the frustum is h centimeters, find the height of the cone


 











Using similar triangle
                       

Question 44



 










In the diagram above, the base diameter is 14cm, whilke the height is 12cm. Calculate the total surface   area if the cylinder has bothe a base and a top [p =22/7] (UME 1995, Question 33)

Solution

Question 46






In the diagram above, find PQ, If the area of the triangle PQR is 35cm2 (UME 1995, Question 34)

Solution

Area of triangle =
PQ  =14cm

 

Question 47







The figure above shows circles radii  3cm and 2cm with center at X and Y respectively. The circle have a transverse common tangent of length 25cm. Calculate XY (UME 1997 Question 27)

Solution









Question 48
A cone with the sector angle of 45o is cut out of a circle of radius r cm. Find the base radius of the cone. (UME 1997, Question 31)

 

 

 

Solution




 










Question 49










In the figure above PQST is a parallelogram and TSR is a straight line. If the area of   QRS  is 20cm2. Find the area of trapezium PQRT

Solution

 

 

Question 50

A cylindrical drum of diameter 56cm contain 123.2 litres of oil when full. Find the height of the drum in centimeters

Solution


Question 51



 








Find the value of l  in the diagram above (UME 1999, Question 32)

Solution

















Question 52
A cylindrical tank has a capacity of 3080m3. What is the depth of the tank  if the diameter of its base is 14m (UME 2001, Question 23)

Solution


Question 53


 









In the diagram above are two concentric circles of radii r and R respectivelywith centre O. If .     Express the area of the shaded portion in terms of

Solution

Question 54
A bucket is 12cm in diameter at the top,  8cm in diameter at the bottom and 4cm deep. Calculate its volume

 

 

 

Solution










Question 55
 











In the diagram above, a cylinder is surmounted by a hemispherical bowl. Calculate the volume of the solid

Solution


Question 56
A trapezium has two parallel sides of length 5cm and 9cm. If the area is 21cm2. Find the distance between the parallel sides  (UME 2003, Question 26)

Solution







Question 57









In the  diagram above, PQ = 4cm  and TS = 6cm , if the area of parallelogram PQTU is 32 cm2, Find the          area of trapezium PQRU

Solution 58









Area of parallelogram  = bh


 

Question 59



In the diagram above  is the diameter of the semicircle QR. Find the area of the figure to the nearest whole number (UME 2006, Question 27)

Solution


Question 60

The volume of a hemispherical bowl is . Find its radius.

Solution


Question 61
 









Find the area of figure above (a) 12.5cm2 (b) 75.0cm2  (c) 78.5cm2 (d)  8.8cm2
(UME 2008, Question 28, Set A U03)

Solution










Area of rectangle ACDE =
Area of semicircle =
Area of the figure =
Answer: Option D

Question 62
Find the capacity in litres of a cylindrical well of radius 1metre and depth 14metres
(a)  44,000litres  (b)  4400litres  (c)  440litres  (d)  44litres
(UME 2008, Question 29, Set A U03)


Solution
Given r =1 metre,  depth = h =14 metres
Volume of cylinder =   
Note: 1m3 =1 litre
44m3 = 44 litres
Answer: Option D


            Question 63
            Find the radius of a sphere whose surface area is 154cm2, take
            (a)   7.00cm   (b)  3.50cm   (c) 3.00cm (d) 1.75cm 
(UME 2009, Question 29, Set A U03)

Solution
             
            Answer: Option B

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