Appen I Modeling Problems

WEEK 5 Problem-based learning Direct assessment method

APPENDIX I

There are TWENTY questions altogether. You are required to answer FIVE questions, selecting one from each color code. This is a teamwork assessment and is due October 5 (week 12). Each of you will be evaluated by your peers. No late assignments will be accepted. No. 1 2.

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Question The population of a certain community that is known to increase at a rate proportional to the number of people present at any time is 10000 after 3 years. What was the initial population? What will be the population in 10 years? The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at any time. After 3 hours it is observed that there are 400 bacteria present After 10 hours there are 2000 bacteria present. What is the initial number of bacteria? Five college students with flu return to an isolated campus of 2500 students. If the rate at which this virus spreads is proportional to the number of infected students y as well as the number not infected 2500 − y , solve the initial value problem

dy = ky ( 2500 − y ) , y ( 0 ) = 5 to find the number of infected students after t days if 25 dt 4. 5.

students have the virus after one day. How many students have the flu after five days? Initially there were 100 milligrams of radioactive substance present. After 6 hours the mass decreased by 35. If the rate of decay is proportional to the amount of the substance present at any time, find the amount remaining after 24 hours. Show that the half-life of a radioactive substance is, in general t =

A1 = A( t1 ) and A2 = A( t 2 ) , t1 < t 2

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( t 2 − t 2 ) ln 2 ln( A1 / A2 )

where

A thermometer is removed from a room where the air temperature is 70° F to the outside where the temperature is 10° F . After ½ minute the thermometer reads 50° F . What is the reading at t = 1 minute? How long will it take for the thermometer to reach 15° F ? The formula

dT = k ( T − Tm ) where k is a constant of proportionality also holds when dt

an object absorbs heat from the surrounding medium. If a small metal bar whose initial temperature is 20°C is dropped into a container of boiling water, how long will it take the bar to reach 90°C if it is known that its temperature increased 2° in 1 second? How long will it take the bar to reach 98°C ? Determine the time of death if a corpse is 79oF when discovered at 3.00 P.M. and 68oF three hours later. Assume that the temperature of the surroundings is 60oF and the normal body temperature is 98.6oF. A can of diet cola at room temperature of 70oF is placed in a cooler with temperature 40oF. After 30 minutes, the can is 60oF. When is the can 45oF? When a cup of coffee is poured its temperature is 200oF. Two minutes later, its temperature is 170oF. If the temperature of the room is 68oF, when is the temperature of the coffee 140oF? Find the family of orthogonal trajectories to 4 x − y = c. Find the family of orthogonal trajectories to y − x 2 = c. Find the family of orthogonal trajectories to y = x − 1 + ce − x .

Find the family of orthogonal trajectories to y (1 − cx ) = 1 + cx .

Puan Azizan Zainal Abidin EBB1113 Differential Equations

July 2009 semester

WEEK 5 Problem-based learning Direct assessment method

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APPENDIX I

A tank contains 200 liters of fluid in which 30g of salt is dissolved. Brine containing 1 g of salt per liter is then pumped into the tank at a rate of 4 liters per minute; the wellmixed solution is pumped out at the same rate. Find the number of grams of salt A(t ) in the tank at any time. A tank contains 200 liters of fluid in which 30g of salt is dissolved. Pure water is then pumped into the tank at a rate of 4 liters per minute; the well-mixed solution is pumped out at the same rate. Find the number of grams of salt A(t ) in the tank at any time. A large tank is filled with 500 gallons of pure water. Brine containing 2lb of salt per gallon is pumped into the tank at a rate of 5 gallons per minute. The well-mixed solution is pumped out at the same rate. Find the number of pounds of salt A(t ) in the tank at any time. The rate at which a drug disseminates into the bloodstream is governed by the differential

dy = A − BX where A and B are positive constants. The function X (t ) dx describes the concentration of the drug in the bloodstream at any time t . Find the limiting value of X as t → ∞ . At what time is the concentration one-half this limiting value? Assume that X (0) = 0. equation

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When forgetfulness is taken into account, the rate of memorization of a subject is given

dA = k1 ( M − A) − k 2 A where k1 > 0, k 2 > 0, A(t ) is the amount of material dt memorized in time t. M is the total amount to be memorized, and M − A is the amount remaining to be memorized. Solve for A(t ) and graph the solution. Assume that 70oF. Find the limiting value of A as t → ∞ and interpret the result. Suppose that glucose enters the bloodstream at the constant rate of r grams per minute while it is removed at a rate proportional to the amount y present at any time. Solve the dy = r − ky, y ( 0 ) = y 0 to find y ( t ) . What is the eventual initial-value problem dt by

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concentration of glucose in the bloodstream according to this model?

Puan Azizan Zainal Abidin EBB1113 Differential Equations

July 2009 semester