11. A bacteria culture is known to grow at a rate proportional to the amount present. After one hour, 1000 strands of the bacteria are observed in the culture; and after four hours, 3000 strands. Find (a) an expression for the approximate number of strands of the bacteria present in the culture at any time t and (b) the approximate number of strands of the bacteria originally in the culture.

12. In 1990 the Department of Natural Resources released 1000 splake (a crossbreed of fish) into a lake. In 1997 the population of splake in the lake was estimated to be 3000. Estimate the population of splake in the lake in the year 2020.

13. Therateofchangeofyisproportionaltoy.Whent=0,y=3,andwhent=2,y=6.Whatis thevalueofywhent=3?

14. Bacteriainalabculturegrowsinsuchawaythattheinstantaneousrateofchangeofbacteriaisdirectlyproportionaltothenumberofbacteriapresent.A.)Writeadifferentialequationthatexpressestherelationship.B.) Solvethedifferentialequationforthenumberofbacteriaasafunctionoftime

15. Suppose that a population of fruit flies grows exponentially. If there were 100 flies after the second day and 300 flies after the fourth day, how many flies were in the original population?*Hint: C=100e-2k, substiture this value and solve for k first.

16. Supposethatinitiallythereare3millionbacteriaand2hourslaterthatnumberhasgrownto5million.Writetheparticularequationthatexpressesthenumberofmillionsofbacteriaasafunctionofhours. Whenwillthepopulationsamplereach1billion?

17. In 1980 the population of alligators on the Kennedy Space Center grounds was estimated to be 1500. In 2006 the population had grown to an estimated 6000. Using the Malthusian law for population growth, estimate the alligator population on the Kennedy Space Center grounds in the year 2020.

18. Bacteria in a culture increased from 400 to 1600 in three hours. Assuming that the rate of increase is directly proportional to the population, a) Find an appropriate equation to model the population (assuming P0 = 400 at time t = 0). b) Find the number of bacteria at the end of six hours (t = 6) using the equation found above.

19. A colony of bacteria with originally 100 cells has now grown to 400 in 2 hours. Assuming the rate of growth is proportional to the present number, nd the number of bacteria in 5 hours from now

20. A certain population grows exponentially. The population grows from 3500 people to 6245 people in 8 years. How long will it take for the original population to double?

11. Radioactiveradiumhasa halflifeofapproximately1599years. Whatpercentofagivenamountremainsafter100years?

12. The half-life of plutonium PU-239 is 24,360. If 10 grams of plutonium was released during the Chernobyl nuclear accident, how long will it take to decay to only 1 gram?a.) Determine points that could be used to solve the problemb.) Create a model using y=Cekzc.) Use the model to determine the value when 1 gram is remaining

13. A manufacturing plant notices its sales dropped from 100000 units per month to 80000 units per month. If the sales follow an exponential pattern of decline, what will they be after 2 months?

14. This year 500 pounds of Plutonium 239 (nuclear waste) was dumped in the Nevada desert. The half life of Pu-239 is 24,360 years. How long will it take until there is only 1 pound left?

15. Four months after it stops advertising, a manufacturing company notices that its sales have dropped from 100,000 units per month to 80,000 units per month. If the sales follow an exponential pattern of decline, what will they be after another 2 months?

16. If initially there are 300 g of a radioactive substance and after 5 yr there are 200 g remaining, how much time must elapse before only 10 g remain?

17. Bacteria in a culture increased from 400 to 1600 in three hours. Assuming that the rate of increase is directly proportional to the population,a) Find an appropriate equation to model the population (assuming P0 = 400 at time t = 0).b) Find the number of bacteria at the end of six hours (t = 6) using the equation found above.

18. Living tissue contains two isotopes of carbon, one radioactive and the other stable. (The ratio of the two being constant). But the radioactive one decays with a half-life of about 5500 years. Find k in y=Cekt

19. You nd a skull in a nearby Native American ancient burial site and with the help of a spectrometer, discover that the skull contains 9% of the C-14 found in a modern skull.Assuming that the half life of C-14 (radiocarbon) is 5730 years, how old is the skull?

20. The charcoal from a tree killed in the volcanic eruption that formed that formed Crater Lake in Oregon contained 44.5% of the carbon-14 found in living matter. About how old is Crater Lake?

1. Suppose a 120 gallon well-mixed tank initially contains 90 lb. of salt mixed with 90 gal. of water. Salt water (with a concentration of 2 lb/gal) comes into the tank at a rate of 4gal/min. The solution flows out of the tank at a rate of 3 gal/min. How much salt is in the tank when it is full?

2. A full 20 liter tank has 20 grams of yellow food coloring dissolved in it. If a yellow food coloring solution (with a concentration of 2 grams/liter) is piped into the tank at a rate of 3 liters/minute while the well mixed solution is drained out of the tank at a rate of 3 liters/minute, what is the limiting concentration of yellow food coloring solution in the tank?

3. Let represent the temperature of an object in a room whose temperature is kept at a constant 60 degrees. If the object cools from 100 degrees to 90 in 10 minutes, how much longer will it take for its temperature to decrease to 80 degrees?

4. A 150 gallon tank has 60 gallons of water with 5 pounds of salt dissolved in it. Water with a concentration of 2 + cos(t) lbs/gal comes into the tank at a rate of 9 gal/hour. If the well mixed solution leaves the tank at a rate of 6 gal/hour, how much salt is in the tank when it overflows?

5. A brine solution of salt flows at a constant rate of 6 L/min into a large tank that initially held 50 L of brine solution in which was dissolved 0.5 kg of salt. The solution inside the tank is kept well stirred and flows out of the tank at the same rate. If the concentration of salt in the brine entering the tank is 0.05 kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.03 kg/L?

6. A brine solution of salt flows at a constant rate of 4 L/min into a large tank that initially held 100 L of pure water. The solution inside the tank is kept well stirred and flows out of the tank at a rate of 3 L/min. If the concentration of salt in the brine entering the tank is 0.2 kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.1 kg/L?

7. A pot of liquid is put on the stove to boil. The temperature of the liquid reaches 170oF and then the pot is taken off the burner and placed on a counter in the kitchen. The temperature of the air in the kitchen is 76oF. After two minutes the temperature of the liquid in the pot is 123oF. How long before the temperature of the liquid in the pot will be 84oF?

8. We start with a tank containing 50 gallons of salt water with the salt concentration being 2 lb/gal. Salt water with a salt concentration of 3 lb/gal is then poured into the top of the tank at the rate of 3 gal/min and salt water is at the same time drained from the bottom of the tank at the rate of 3 gal/min. We will consider the water and salt mixture in the tank to be well-stirred and at all times to have a uniform concentration of salt. Find the function S(t) that gives the amount of salt in the tank as a function of time (t) since we began pouring in salt water at the top and simultaneously draining salt water from the bottom of the tank. How long before there will be 120 pounds of salt in the tank?

9. A cold beer initially at 35F warms up to 40F in 3 min while sitting in a room of temperature 70F.How warm will the beer be if left out for 20 min?

10. A red wine is brought up from the wine cellar, which is a cool 10C, and left to breathe in a room of temperature 23C. If it takes 10 min for the wine to reach 15C, when will the temperature of the wine reach 18C?

11. An object of mass 3 kg is released from rest 500 m above the ground and allowed to fall under the influence of gravity. Assume the gravitational force is constant, with g = 9.81 m/sec2, and the force due to air resistance is proportional to the velocity of the object with proportionality constant b = 3 N-sec/m. Determine when the object will strike the ground.

12. A 400-lb object is released from rest 500 ft above the ground and allowed to fall under the influence of gravity. Assuming that the force in pounds due to air resistance is _10y, where y is the velocity of the object in ft/sec, determine the equation of motion of the object. When will the object hit the ground?

13. An object of mass 8 kg is given an upward initial velocity of 20 m/sec and then allowed to fall under the influence of gravity. Assume that the force in newtons due to air resistance is -16y, where y is the velocity of the object in m/sec. Determine the equation of motion of the object. If the object is initially 100 m above the ground, determine when the object will strike the ground.

14. A shell of mass 2 kg is shot upward with an initial velocity of 200 m/sec. The magnitude of the force on the shell due to air resistance is [v]/20. When will the shell reach its maximum height above the ground?