A tank contains 10 liters of pure water. A solution of an unknown, but constant concentration of salt is flowing in at 0.5 liters per minute. The water is mixed well and drained at 1 liter per minute. After 20 minutes there are 10 grams of salt in the tank. (i) Give an adequate mathematical model for this scenario. Explain what each of your variables means. (ii) What is the concentration of the incoming salt solution

i) ∴∫  = ∫ ------ ( 1 )

ii) C = 4.303

Explanation:

Given data:

water in tank = 10 liters

concentration of salt flowing in = 0.5 liters/min

10 grams of salt is left in tank after 20 minutes

mixed water is drained at 1 liter per minute

i) Adequate mathematical model for the scenario

lets assume ; A( t ) amount of salt after t minute

A( t ) = 10grams   where t = 20 minutes

differentiate A(t)

dA / dt = ( rate in ) - ( rate out )

           = 0.5 * 0  - A/10

∴∫  = ∫ ------ ( 1 )

ii) hence;  In A =  -----  ( 2 )

at  t = 20 , A = 10grams

find C = In 10 + 2

          = 2.303 + 2 = 4.303

back to ( 2 )

In A =

∴ A = e

answer : a model is often used to explain and make things easier to understand when any scientific topic needs to be demonstrated and explained.

as in this case where a scientist wants to use a model to present the results of his detailed scientific investigation. he should do it as it will him to make the audience understand the concept in a much better way. also, it can be changed according to new scientific understandings.