Exploring Salt Concentration Changes in a Mixing Vat

How does the mass of salt in a mixing vat change over time?

Which of the following is an explicit expression for m(t)?

A. m(t) = 900-900e^(t/60)

B. m(t) = 900 + 900e^(t/60)

C. m(t) = 900-900e^(-t/60)

D. m(t)= 900+ 900e^(-t/60)

E. m(t) = 900e^(-t/60)

Answer:

The explicit expression for m(t), the mass of salt present in the mixing vat after t minutes, is given by option C: m(t) = 900 - 900e^(-t/60).

In this scenario, the mass of salt in a mixing vat changes dynamically as brine solution is pumped in while the mixed solution is being pumped out simultaneously. Initially, the vat contains only pure water.

The rate at which salt is entering the vat is calculated based on the concentration of salt in the brine solution, which is 3 g/L. Meanwhile, the rate of salt being pumped out depends on the salt concentration within the vat.

By setting up a differential equation to represent the rate of change of salt mass with time, we can solve for the explicit expression of the mass of salt in the vat at any given time using initial conditions.

The final explicit expression for m(t) is m(t) = 900 - 900e^(-t/60), showcasing the dynamic nature of salt concentration changes in the mixing vat.

For further insights on mass-related topics, you can explore more here.