Question

(1) In rectangles with one side l centimetre shorter than the other, take the length of the shorter side as x centimetres.

I) Taking their perimeters as p(x) centimetres, write the relation between p(x) and .x as an equation.

ii) Taking their areas as a(x) square centimetres, write the relation between a(x) and x as an equation.

iii)Calculate p(1), p(2). p(3), p(4), p(5). Do you see any pattern?

iv) Calculate a(1), a(2), a(3), a(4), a(5) Do you see any pattern?​

Answer 1

Given,

       Let,

•    Length of rectangle = x cm
•    Breadth of rectangle = x-l cm

Therefore,

1.   Perimeter of rectangle = 2 ( x + (x-l) )

           Perimeter of rectangle = 2 ( 2x - l )

           Perimeter of rectangle = 4x - 2l

                     Its equation,

                     $\boxed{p(x) = 4x - 2l}$

   2.   Area of rectangle = l × b

         Area of rectangle = x × (x-l)

         Area of rectangle = x² - xl

                     Its equation,

                     $\boxed{a(x) = x^2 - lx}$

   3.   Equation of p(x) = 4x - 2l

          Therefore,

•   p(1)  = 4x - 2l = 4(1) - 2l = 4 - 2l
•   p(2) = 4x - 2l = 4(2) - 2l = 8 - 2l
•   p(3) = 4x - 2l = 4(3) - 2l = 16 - 2l
•   p(4) = 4x - 2l = 4(4) - 2l = 20 - 2l
•   p(5) = 4x - 2l = 4(5) - 2l = 24 - 2l

       Yes, there is a pattern. Its nth term is  $a_{n}$ =  4(n) + 2l

       where n ∈ z

       

 4.   Equation of a(x) = x² - lx

        Therefore,

•  a(1) = x² - lx = 1² - l(1) = 1 - l
•  a(2) = x² - lx = 2² - l(2) = 4 - 2l
•  a(3) = x² - lx = 3² - l(3) = 9 - 3l
•  a(4) = x² - lx = 4² - l(4) = 16 - 4l
•  a(5) = x² - lx = 5² - l(5) = 25 - 5l

       Yes, there is a pattern. Its nth term is  $a_{n}$ = n² × (n)l

        Where n ∈ z

       

Answer 2

Given :   In rectangles with one side l centimetre shorter than the other, take the length of the shorter side as x centimetres.

To Find : Taking their perimeters as p(x) centimetres, write the relation between p(x) and .x as an equation.

Solution:

length of the shorter side = x   cm

one side l cm shorter than the other,

=> x  = other side - l

=> other side = x + l cm

Perimeter = 2( x + x + l )  =  2 ( 2x + l)  = 4x  + 2l  cm

p(x) = 4x  + 2l  

Area = x ( x + l)  = x² + lx

=> A(x) =  x² + lx

P (1)  =  4 + 2l  

P(2) = 8 + 2l

P(3) = 12 + 2l

P(4) = 16 + 2l

P(5) = 20 + 2l

Perimeter increase by 4 with increase of one in value of x

A(1)  = 1 +  l

A(2) = 4 + 2l

A(3) = 9 + 3l

A(4) = 16 + 4l

A(5) = 25 + 5l  

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