In rectangles with one side 1 centimetre shorter than the other, take

1.	In rectangles with one side 1 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» Let x be the shorter side, then the other side will be (x + 1). i. Perimeter = 2[x + (x + 1)] = 2(2x + 1) = 4x + 2 That is, p(x) = 4x + 2 ii. Area = x(x + 1) a(x) = x² + x Area, a(x) = x² + x iii. p(x) = 4x + 2 p(1) = 4 × 1 + 2 = 6 p(2) = 4 × 2 + 2 = 10 p(3) = 4 × 3 + 2= 14 p(4) = 4 × 4 + 2= 18 p(5) = 4 × 5 + 2 = 22 Perimeter is a sequence increasing by 4. iv. a(x) = x² + x a(1) = 1² + 1 = 2 = 1 × 2 = 2 a(2) = 2² + 2 = 6 = 2 × 3 = 6 a(3) = 3² + 3 = 12 = 3 × 4 = 12 a(4) = 4² + 4 = 20 = 4 × 5 = 20 a(5) = 5² + 5 = 30 = 5 × 6 = 30 Area is the product of x and the number one more than x.

In rectangles with one side 1 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»

Let x be the shorter side, then the other side will be (x + 1).

i. Perimeter = 2[x + (x + 1)] = 2(2x + 1) = 4x + 2

That is, p(x) = 4x + 2

ii. Area = x(x + 1)

a(x) = x² + x

Area, a(x) = x² + x

iii. p(x) = 4x + 2

p(1) = 4 × 1 + 2 = 6

p(2) = 4 × 2 + 2 = 10

p(3) = 4 × 3 + 2= 14

p(4) = 4 × 4 + 2= 18

p(5) = 4 × 5 + 2 = 22

Perimeter is a sequence increasing by 4.

iv. a(x) = x² + x

a(1) = 1² + 1 = 2 = 1 × 2 = 2

a(2) = 2² + 2 = 6 = 2 × 3 = 6

a(3) = 3² + 3 = 12 = 3 × 4 = 12

a(4) = 4² + 4 = 20 = 4 × 5 = 20

a(5) = 5² + 5 = 30 = 5 × 6 = 30

Area is the product of x and the number one more than x.