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.

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

Two poles of heights $3 \mathrm{m}$ and $4 \mathrm{m}$ are erected upright on the ground, $5 \mathrm{m}$ apart. A rope is to be stretched from the top of one pole to some point on the ground and from there to the top of the other pole:

The relation between the distance of the point on the ground from the foot of a pole and the total length of the rope.