Question

Water tank is being emptied in such a way that the rate at which
water is flowing out is proportional to the amount left in at that
instant. If half the water flows out in 7 minutes. Initially the tank is
filled with 8000 liters of water. Find the amount of water left after
21 minutes.​

Answer 1

Volume of water tank = 5m²

We know 1m³

=1000L

then,

volume of water tank in liter = 5000 L.

Answer 2

Answer:

The amount of water left after 21 minutes is 1000 litres.

Step-by-step explanation:

Step 1:

It is given,

$rate \ of\ water\ flowing\ out \ \propto \ Quantity\ of\ water\ present\ in\ the\ tank\ at\ that\ moment\ (Q)$

Mathematically we can write,

                    $\frac{-dQ}{dt} \propto Q$

-ve sign indicates water flowing out.

To make the proportionality into equality we multiply a proportionality constant 'k'.

                   $\frac{dQ}{dt} =- kQ$

Step 2:

Integrating the above differential equation, we get,

                  $\int {\frac{1}{Q} } \, dQ = \int {-k} \, dt \\\\\\ ln(Q) = -kt+c$---------------- (1)

'c' is the integration constant.

Step 3:

It is given initially the tank is filled with 8000 litres of water, implies at t=0, Q=8000=$Q_{i}$, Substituting in equation (1), we have,

                $ln(8000) = -k(0)+c\\c=-ln(Q_{i})$

Substituting for 'c' and rearranging the equation,

                $\frac{Q}{Q_i} = e^{(-kt)}$

Rearranging to get 'k'

                $k=\frac{1}{t} ln(\frac{Q_{i} }{Q})$

Step 4:

To get the final equation for Q we have to find the value of 'k'. It is also given that at t=7 minutes, the tank is half filled, this implies

              $at \ t=7,\ Q=\frac{Q_i}{2} \\k=\frac{1}{7} ln(\frac{Q_{i} }{Q_i/2})\\k=\frac{1}{7}ln(2)$

The final equation for 'Q' is,

              $\frac{Q}{Q_i} = e^{(-\frac{ln2}{7} t)}$

Step 5:

We are required to find the amount of water left after 21 minutes, that is to find Q for t=21.

              $\frac{Q}{Q_i} = e^{(-\frac{ln2}{7} 21)}\\\frac{Q}{8000} = \frac{1}{8}\\ Q=1000$

Therefore the amount of water left after 21 minutes is 1000 litres.