In rectangle with one side $1 \mathrm{cm}$ shorter than the other, take the length of the shorter side as $x \mathrm{cm}$. If the area of the rectangle denotes as $a\left(x\right)$, then calculate $a\left(1\right), a\left(2\right), a\left(3\right), a\left(4\right), a\left(5\right)$. Do you see any pattern?

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:

Taking a side of the square as $x \mathrm{cm}$, write the dimension of the box in terms of $x$.

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:

Taking the volume of the box as $v\left(x\right) {\mathrm{cm}}^3$, write the relation between $v\left(x\right)$ and $x$ as an equation.

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:

If the volume of the box denotes as $v\left(x\right)$, then calculate $v\left(\frac{1}{2}\right), v\left(1\right), v\left(1\frac{1}{2}\right)$.

Consider all rectangles that can be made with a $1 \mathrm{m}$ long rope. Take one of its sides as $x \mathrm{cm}$ and the area enclosed as $a\left(x\right) {\mathrm{cm}}^2$. Write the relation between $a\left(x\right)$ and $x$ as an equation.

Consider all rectangles that can be made with a $1 \mathrm{m}$ long rope. Take one of its sides as $x \mathrm{cm}$ and the area enclosed as $a\left(x\right) {\mathrm{cm}}^2$. Why are the numbers $a\left(10\right)$ and $a\left(40\right)$ equal?

Consider all rectangles that can be made with a $1 \mathrm{m}$ long rope. Take one of its sides as $x \mathrm{cm}$ and the area enclosed as $a\left(x\right) {\mathrm{cm}}^2$. To get the same number as $a\left(x\right)$, for two different numbers $x$, what must be the relation between the numbers?

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

An $1 \mathrm{m}$ wide path goes around a square ground. The relation between the length of a side of the ground and the area of the path.

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

A liquid contains $7$ litres of water and $18$ litres of acid. More acid is added to it. The relation between the amount of acid added and the change in the percentage of acid in the liquid.