Q3 of 16 Page 147

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 numbers as a(x), for two different numbers as x, what must be the relation between the numbers?

According to the question the perimeter of the rectangle must be always equal to 100 cm.


Now, let one side be x cm and other adjacent side be y cm.


⇒ P = 2 × [x + y]


⇒ 100 = 2x + 2y


⇒ y = 50 – x – (1)


(i) a(x) = (50-x) × (x) – (by (1))


a(x) = 50x – x2


(ii) a(10) = 50 × (10) – (10)2 = 400


a(40) = 50 × (40) – (40)2 = 400


Now if a side a is 10 cm, then the other side


P = (10 + y) × 2


P = 100


⇒ 50 – 10 = y


⇒ y = 40 cm


∴ the in case of x = 10 cm y = 40 cm, and in case of x = 40 cm y = 10 cm. So, in both cases we have same rectangle hence the area is same.


More from this chapter

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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 area 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?

 2

From the four corners of a rectangle, small squares are cut off and the sides are folded up to make a box, as shown below:


i) Taking side of the square as x centimetres, write the dimensions of the box in terms of x.


ii) Taking the volume of the box as v(x) cubic centimetres, write the relation between v(x) and x as an equation.


iii) Calculate

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 1

Write each of the relations below in algebra and see if it gives a polynomial. Give reasons for your conclusion also.

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