Question

Consider all rectangles that can be made with a 1-metre long rope. Take one of its sides as x centimetres and the area enclosed as a(x) square centimetres.

i. Write the relation between a(x) and x as an equation.

ii. Why are the numbers a(10) and a(40) equal?

iii. To get the same number as a(x), for two different numbers as x, what must be the relation between the numbers?

Answer 1

i. 1 m = 100 cm

If one side is x cm, then the other side is 50 – x.

Area of rectangle = a(x) = x(50 – x)

= 50x – x² cm²

a(x) = 50x – x²

ii. a(10) = 50 × 10 – 10² = 500 – 100 = 400

a(40) = 50 × 40 – 40² = 2000 – 1600 = 400 

10 and 40 are the sides of rectangle, so a(10) and a(40) are same .

iii. Numbers must be sides of rectangle. If we add the two numbers together we get the sum as half the length of the wire (50 cm) used to make it.