Question:Find the radios of a circle of area 38.5 `cm^2`  (`pi =22/7` ). 

`pi r^2` = 38.5

`:. r^2` = `38.5/pi` 

         = `(38.5 xx 7)/22`

         = `(3.5 xx 7)/2`

         = `(3.5)^2` .

   `:.`r = 3.5

The radius of the circle is 3.5 cm

Question:Find the area between two circles of radii 4 cm and 3 cm . given that the smaller circle lies entirely within the larger. 

Area of the bigger circle =`pi(4)^2 cm^2` =16`pi cm^2`.

Area of the smaller circle=`pi(3)^2``cm^2`= 9`pi cm^2` .

Area between the circle = (16`pi`- 9`pi`) `cm^2`= 7`pi cm^2`.

                                      = 7`xx 22/7 cm^2`

                                      = 22 `cm^2`

Question:Find  the volume and total surface  area of a right  circular cylinder , the radius of whose base  is 4 cm and whose height is 5 cm  ( Take `pi` to be 3.142) . 

The volume =`pi r^2`h=`pi(4)^2 5 cm^3`

                   = 80`pi cm^3`.

                   = 251.36 `cm^3`

                   = 251 `cm^3`

 The area of the curved  surface = 2`pi`rh

                                 = 2`pi`(4)(5) `cm^2`  

                                 = 40`pi cm^2`.

The area of each end = `pir^2`=`pi(4)^2 cm^2` = 16`pi cm^2`.

The total surface area =(40`pi`+32`pi`) `cm^2`

                                   = 72`pi cm^2`

                                   =226.224 `cm^2`

                                   226 `cm^2`

Question:Find the volume of material in pipe of length 7 cm, if the internal and external radii are 3 cm and 4 cm (Take `pi` to be `22/7`). 

The volume of the larger cylinder

                   = `pi(4)^2` 7 c`m^3`= `pi xx16 xx 7 cm^3`= 112`pi cm^3`.

The volume of the smaller cylinder

                   = `pi(3)^2` 7 c`m^3`= 63`pi cm^3`

The volume of the material
                   = (`112 pi - 63 pi` ) `cm^3`

                   = 49`pi cm^3`

                   = 49`xx 22/7 cm^3`.

                   = 154 `cm^3`.

Question:Evaluate 7`xx`19.3+8`xx`19.3-5`xx`19.3 

7`xx`19.3+8 `xx` 19.3 - 5`xx`19.3
 =19.3(7+8-5)
 =19.3 `xx`10
 =193

Question:Evaluate 19.3`pi` - 15.8`pi` 

19.3`pi` - 15.8`pi` =`pi`(19.3-15.8)=`pi` `xx`3.5 

                                   = `22/7 xx 7/2`

                                   = 11

Question:Factories ax+bx+ay+by 

ax+bx+ay+by = (ax+bx) + (ay+by)
                       = x(a+b) +y (a+b)
                       =(a+b) (x+y).

Question:Factorise  ax + ay -x -y . 

ax + ay -x -y=a(x+y) - (x+y) 
                    =(x+y) (a-1).

 Notice that when the last two terms are grouped with a minus sign outside. the signs of both terms in the bracket must be changed.

                    ax + ay -x -y=a(x+y) - (x+y) 
 and not     
                   a(x+y) - (x-y) .

It is good plan to remove the brackets mentally to check your working.

Question:Factorise `xx^2`y - `xxy^2`+3y -3`xx` 

`xx^2`y - `xxy^2`+(3y -3`xx`) = xy(x-y) +3(y-x)

This has no obvious factor as the expression inside the brackets are different .
  but if we arrange the expression as 
   
`xx^2`y - `xxy^2`-3`xx`+3y) = `xx`y(x-y) -3(x-y),

An obvious factor is now apparent.
The factors are (x-y)(xy-3).

Question:Factorise `xx^2`+9x+20. 

`xx^2`+9x+20

=`xx^2`+5x +4x +20

=x(x+5)+(x+4)

=(x+5)  (x+4)