12.A certain polluted lake holds 50kL of water. Pure water flows into the lake at arate of 2kL per hour, and water flows out of the lake at the same rate. The lake is polluted withcyanide, the initial concentration of which is 3g per kL. In order to be deemed safe to swim in, theconcentration of cyanide has to decrease to 1g per kL. Assuming instantaneous mixing, how longwill it be before the water is safe to swim in?25 ln 3 hours

Lets say amount of cyanide at time 't' be x(t) gram The rate of change of amount of cyanide is…

Answered: 12. A certain polluted lake holds 50kL…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1. Mr. Pein has an aquarium that has a volume of 2000 mL and contains freshwater fish. One day, Mr. Pein's son accidentally spilled 1 kilogram of salt into the aquarium. The fish will die if they are in water with a salt concentration of more than 10-3 grams/mL for more than 30 minutes. So that the fish in the aquarium do not die, Mr. Pein must immediately reduce the salt concentration in the aquarium to 10-3 grams/mL by flowing fresh water at a constant rate through the aquarium, and removing the water from the aquarium at the same rate. What is the minimum water speed so that the fish don't die? In 500 А. 60000 mL/minute 2000 В. 30 In 500 mL/min 1996 С. mL/minute 30 In 500 D. 6000 mL/minute 200 Е. 30 In 500 mL/min
9. A patient weighing 70 kg should receive a dose of 1400 units of rabies immune globulin after exposure to rabies. How much of a solution containing 900 units per 3 mL should be injected for this dose? 10. Zoster Vaccine Live (ZOSTAVAX) contains 19,400 plaque-forming units (PFU) of attenuated virus per 0.65 mL. What is the concentration expressed as PFU/mL?
11. A tank contains 40 gal beine for which the concentration is initialy 3 b/gal of salt. A brine soluion is pumped into the fonk at the rote of 5 gal/min and the mixture. kept uniform by stiring, runs out at the same rate. the concentration of the solution flowing in a 2 b/gal a) determine the amount of salt in the tank at anytime t b) how much salt is present offer 10 min? e) affer a long time? 12. A tank contoins 80 gal of pure woter. Brine solution with 2 b/gol of solt enters at 2 gat/min, and the well-stimed mlature leaves at the same rate. Find a) the amount of salt in the tank after hal an hour? b) determine the imiting value for he amount of solt in the tank offer a long fime, and e) how much time must pass before the amount of saltl in the tank reaches 80% of this imiting value? 13. A certain type of glass is such that a slab l" thick absorbs one-quarter of the light which storts to pass through it. How thin must a pane of be made to absorb only 15 of the light? 14. Find the…
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: (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 gallons minutes is V(t)=500-5r gallons (c) The volume of the solution…
9. A glass container can hold 35 liters of water. It currently has 10 liters of water with 15 grams of Gatorade power initially dissolved in the container. A solution is poured into the container at 3 liters per minute - the solution being poured in has 0.5 grams per liter of Gatorade powder. Assume the solution in the container is well mixed. There is an outflow at the bottom of the container which has liquid leaving at 1 liter per minute. Let G(t) denote the amount of Gatorade powder in the tank at time t. a. Setup the differential equation for G'(x). b. Solve for the general solutiont) c. Use initial condition to find the specific solution. (Write out the entire solution, with the constant(s) plugged in.) ) d. When will the container overflow?
2. A large tank initially contains 100 gallons of brine in which 10 lb of salt is dissolved. Starting at t = 0, pure water flows into the tank at the rate of 5 gal/min. The mixture is kept uniform by stirring, and the well-stirred mixture simultaneously flows out at the slower rate of 2 gal/min. a) How much salt is in that tank at the end of 15 min and what is the concentration at that time? b) If the capacity of the tank is 250 gallons, what is the concentration at the instant the tank overflows?
5. Suppose that sodium pentobarbital is used to anesthetize a dog. The dog is anesthetized when its bloodstream contains at least 45 milligrams (mg) of sodium pentobarbitol per kilogram of the dog's body weight. Suppose also that sodium pentobarbitol is eliminated from the dog's bloodstream at a rate which is proportional to the amount of sodium pentobarbitol in the bloodstream and that after 5 hours half of the initially prescribed sodium pentobarbitol is elliminated from the bloodstream. Set up and solve an initial value differential equation. What single dose should be administered in order to anesthetize a 50-kg dog for 1 hr?
A tank contains 1800 liters (L) of fluid in which 180 grams (g) of salt is dissolved. Solution containing 20 g/L then pumped into the tank at a rate of 30 L/min; the tank is maintained well mixed and pumped out at the same rate. How many grams of salt are in the tank 20 minutes after the inlet supply is opened? а. 10830 b. 11304 C. 11324 d. 10434 е. 10334
9. Imagine a medieval world. In this world a Queen wants to poison a King, who has a wine keg with 500 L of his favorite wine. The Queen gives a conspirator a liquid containing 5 g/L of poison, which must be poured slowly into the keg at a rate of 0.5 L/min. The poisoner must also remove the well-stirred mixture at the same rate, so that the keg is not suspiciously full. (a) Find a formula for the amount of poison in the keg at any time, measured from the start of the pouring by the poisoner. (b) A plot is hatched for the King to drink wine from the keg while he is on a hunt, where he will become so addled that his prey will surely kill him. The poisoner must pour for a time T, when the poison in the keg reaches a dangerous concentration of 0.005 g/L. Find T. (c) The Lord High Inquisitor of the Realm never learned about differential equations. Nonetheless, knowing the basic numbers (keg size, poison concentrations, etc.), he can produce an estimate for the time T that the poisoner was…
20. An aquarium contains 10 gallons of polluted water. A filter is attached to this aquarium which drains off the polluted water at the rate of 5 gal- lons per hour and replaces it at the same rate by pure water. How long does it take to reduce the pollution to half its initial level?

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