3. A large tank with the capacity of 500 gallons is fully filled with pure water. Brine containing 2 pounds of salt per gallons is pumped into the tank at a rate of 5 gal/min. Find the amount A(t) pounds of salt in the tank at timer minutes, if the well-mixed solution is pumped out (a) with the same rate as the entering one; and (b) with the rate of 10 gal/min. In case (a), how much salt is in the tank after a long time? In case (b), what is the concentration of the solution in the tank by the time it is empty? (c) Let everything be the same except that, initially, the tank is filled with 250 gallons of pure water, while the well-stirred solution is pumped out with the flow rate of 2.5 gal/min. Find (t). What is the concentration of the solution in the tank by the time it is fully filled? Hint: minutes is V(t)=500 gallons (a) The volume of the solution in the tank at time (b) The volume of the solution in the tank at time minutes is V(t) = 500-5r gallons (c) The volume of the solution in the tank at time ſ minutes is V(t) = 250+ gallons Answer: (a) A(z)-1000-1000e after a long time it is 1000 lbs. (b) 4()-1000-10-- (c) A(r) = 5(100+) (100-): the concentration by the time it is empty is 2 lb/gal. the concentration by the time it is fully filled is 1.5 lb/gal. 10 5x10¹ 100+7

(b). Consider 2 intercomected tank. Tank A initially contains 18⁰ L of water and log of salt, and tank B Initially contains looL of water and 25g of salt. The Liquid inside each tank is kept well stirred. Liquid flows from tank A to tank B at a rate of BL/min and liquid flows from tank B to tank A at rate of 34lmin. A salt brine with connectration 179/L of salt flows into tank A at a rate of 4 L/min and the solution drains out at 4 L/min. i). Draw a collection of tanks diagram such that all flow rates, volumes, and the incoming salt concentration are shown on it. ii). Write out a system of first-order linear differential equations that would describle the rates of charge of the concentrations of salt, Cilt), (zlt). Do not forget to write the initial conditions. Do not save the equations.

(2) 8. Two gears, one with 15 teeth and the other one with 20 teeth, fit together as shown in this diagram. When the smaller gear has rotated 20 times, how many times has the larger gear rotated? A tank with a capacity of 5 000 e is being filled by a pipe that delivers water at a rate of 200 ml/s. 9. How long will it take to fill the tank? Give your answer in hours, minutes and seconds. a) If water leaks out of the tank at a rate of 3 ml/s as it is being filled, how long will it now b) take to fill the tank? Give your answer in hours, minutes and seconds. (2) (2)

Assume that T is a linear transformation. Find the standard matrix of T. 7x T: R² →R² first rotates points through radians and then reflects points through the horizontal x₁-axis. A= (Type an integer or simplified fraction for each matrix element. Type exact answers, using radicals as needed.)