Question

Initially fifteen grams of salt are dissolved into 40 liters of water. Brine with concentration of salt 5 grams per liter is added at a rate of 2 liters per minute. The tank is well-mixed and drained at 2 liters per minute. Let x be the amount of salt, in grams, in the solution after t minutes have elapsed. Find a formula for the rate of change in the amount of salt, dx/dt, in terms of the amount of salt in the solution x. Find a formula for the amount of salt, in grams after t minutes have elapsed. How long must the process continue until there are exactly 25 grams of salt in the tank?

Answer 1

The rate of change in the amount of salt, dx/dt, is (2 - 2x/40) grams per minute. The formula for the amount of salt, x, after t minutes is x = (2t - tx/20) + 15 grams. The process must continue until there are exactly 25 grams of salt in the tank, which takes 10 minutes.

Step-by-step explanation:

To find the rate of change in the amount of salt, dx/dt, we need to consider the rate at which salt is added and the rate at which the solution is drained. The rate at which salt is added is 2 grams per minute, because 5 grams per liter is added at a rate of 2 liters per minute. The rate at which the solution is drained is also 2 liters per minute. Therefore, the rate of change in the amount of salt is dx/dt = (2 - 2x/40) grams per minute.

To find the formula for the amount of salt, x, after t minutes have elapsed, we need to integrate the rate of change equation. Integrate dx/dt = (2 - 2x/40) with respect to t to get x = (2t - tx/20) + C, where C is the constant of integration. To find C, plug in the initial condition x = 15 and t = 0. You will get C = 15. Therefore, x = (2t - tx/20) + 15 grams.

To find the time it takes until there are exactly 25 grams of salt in the tank, you need to solve the equation x = 25. Substitute x = 25 into the equation x = (2t - tx/20) + 15 and solve for t. You will get t = 10 minutes.