Question

A tank initially contains 100 gal. of salt solution, where 50 lbs. of salt is added. Salt solution containing 1 lb/gal of salt goes into the tank at the rate of 2 gal/min and the solution thoroughly mixed goes out at the rate of 1 gal/min. Find the amount of pure salt after 100 minutes.

Answer 1

The amount of pure salt after 100 minutes is 100 lbs. In order to find the amount of pure salt after 100 minutes, subtract the amount of salt that leaves the tank from the amount that enters.

Step-by-step explanation:

In order to determine the quantity of pure salt remaining in the tank after 100 minutes, a comprehensive assessment of salt inflow and outflow is necessary.

Initially, the influx of salt into the tank is computed by multiplying the rate of salt solution entry (2 gal/min) by the duration (100 minutes), resulting in 200 gallons of solution.

Given the concentration of salt in the solution as 1 lb/gal, the total salt entering the tank is calculated as 200 * 1 = 200 lbs.

Subsequently, the outflow of solution from the tank, occurring at a rate of 1 gal/min, amounts to 100 gallons over the 100-minute period.

Consequently, the salt content leaving the tank is determined as 100 * 1 = 100 lbs.

To ascertain the quantity of pure salt retained in the tank after 100 minutes, the amount of salt leaving is subtracted from the amount entering, yielding a remaining 100 lbs of pure salt.

This analytical approach provides a systematic method for evaluating the evolving salt concentration in the tank over the specified time frame.

Answer 2

To find the pure salt amount in the tank after 100 minutes, we set up a differential equation that accounts for salt added and the rate of inflow and outflow. We solve the equation with initial conditions to find the amount of salt at a given time.

Step-by-step explanation:

To solve for the amount of pure salt in the tank after 100 minutes, we need to set up a differential equation modeling the situation. Initially, the tank contains 100 gal of salt solution with 50 lbs of salt added to it. Salt solution is added at a rate of 2 gal/min, each gallon containing 1 lb of salt, and the well-mixed solution leaves the tank at a rate of 1 gal/min.

Step-by-step explanation:

1. Let S(t) represent the amount of salt in the tank at time t in minutes.
2. The rate of salt coming into the tank is 2 lbs/min (because 2 gal/min × 1 lb/gal = 2 lbs/min).
3. The rate of salt leaving the tank is S(t) / 100 gal/min, because the solution is well mixed and the tank's volume stays constant at 100 gal (since 2 gal enter for every gallon leaving).
4. The differential equation can be written as dS/dt = 2 - S(t)/100.
5. Solve the differential equation using separating variables or an integrating factor, considering the initial condition S(0) = 50 lbs.
6. Once you have S(t), plug in t = 100 minutes to determine the amount of salt after 100 minutes.

Without solving the actual differential equation here, this explanation provides the steps and setup necessary to reach the solution.