A fixed volume of water is to flow into a rectangular water tank. The rate of flow can be changed by using different pipes. Write the relations between the following quantities as an algebraic equation and in terms of proportions.

i) The rate of water flow and the height of the water level.

ii) The rate of water flow and the time taken to fill the tank.

i). Let the rate of flow be ‘r’ m³/s.

That means, in 1 second, ‘r’ m³ of water will enter the tank.

Let the base area of the tank be ‘a’m².

Then in the 1^st second, that is, when t = 1, the height of water level will be:

[∵ after 1 second, the volume in the tank will be r m³]

In the 2^nd second, that is, when t = 2, the height of water level will be:

[∵ after 2 seconds, the volume in the tank will be 2r m³]

In the 3^rd second, that is, when t = 3, the height of water level will be:

So, we can write:

The height of water after the n^th second = h

⇒

n is a constant because we will put a particular value of n. We want the height of water at that n

a is also constant

So, is a constant.

Thus, we have a relation between two quantities:

Height ‘h’ at the n^th second

Rate of flow ‘r’.

We can write: h = kr.

This is the algebraic equation.

Where k

From the equation, we can see that h is directly proportional to r.

The constant of proportionality is .

ii). We can use the same equation used above, i.e.

The height of water after the n^th second = h

⇒

In this case, h is a constant because of the following two reasons:

1. A fixed volume of water is flowing into the tank.

2. When the tank is filled, it will have a particular value of ‘h’.

n is the number of seconds required to fill the tank. It will change if the rate ‘r’ is increased or decreased.

So, n and r are the variables. Let us bring them to opposite side of the ‘=’ sign:

⇒

⇒

This is the algebraic equation.

‘ah’ is a constant. Let it be ‘k’.

We can write:

So, r is proportional to the reciprocal of n. That means n is inversely proportional to r.

If r increases and ‘n’ decreases, indicating a lesser time sufficient to fill up the tank.

If r decreases and ‘n’ increases, indicating a greater time required to fill up the tank.