8/12/2019 Homework 2 M287

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Math 287 Spring 2014Homework #2 Due Monday 2/31.4 Practice: All True False, 1, 4, 5, 9, 13, 17, 18, 21, 25

1. Solve the equation

2. Find the orthogonal trajectory to the family that passes through the point (1,2)3. A second order chemical reaction starts with substances and and produces substance in such a

way that One molecule of together with one molecule of produces one molecule of , written The rate at which is produced is proportional to the product of the amount of substances and

remaining.Set to be the initial amount of substance and to be the initial amount of substance and let bethe amount of substance at time . Then, at time the amount of substance remaining is , theamount of substance remaining is and so the rate at which is being produced is

. Assuming solve for . Your answer will include the parametersand .

1.5 Practice: All True False, 1, 3, 8, 9, 10, 15, 161. The number of bacteria on the surface of a raw hamburger grows at a rate proportional to the number

present. Initially there were 10,000 bacteria present and 20 minutes later there were 15,000. Find thenumber of bacteria as a function of time

2. Radioactive thorium-234 decays at a rate proportional to the amount present. If 100 mg of this materialis reduced to 82.04 mg in one week how long will it take the 100 mg to be reduced to 30 mg?

3. A rumor spreads at a rate proportional to the product of the number of people who know the rumor andthe number who dont. Suppose in a population of 2,000 people initially 50 have heard the rumor. Findthe number who have heard the rumor as a function of time.

1.6 Practice: All True False, 1, 5, 13, 15, 17, 19, 23 (page 4), 26, 28, 29 1. Solve the differential equation2. Solve the initial value problem3. It is known that a certain object has constant of proportionality in Newtons law of cooling. The

object starts at a temperature of and is placed in a room whose temperature starts at andincreases at the rate of per minute. Find the temperature of the object as a function of time.

1.7 Practice: All True False 1, 4, 8, 9, 14, 15 1. A container initially contains 100 L of water with no salt. A solution containing 4 g/L of salt is pumped into

the container at a rate of 3 L/min, and the well stirred mixture is pumped out of the container at thesame rate. Determine the amount of salt in the container as a function of time.

2. An RL series circuit has a resistance of 30 Ohms, an inductance of 0.4 Henries, and a voltage ofVolts. If no current if flowing initially determine the current as a function of time.

3. Air with a carbon monoxide concentration of 0.001 grams per cubic meter is flowing into a room at therate of 2 per minute. The well mixed air in the room also flows out at the rate of 2 per minute.The room contains a detector that will sound an alarm when the carbon monoxide concentration in theroom reaches 0.0005 grams per cubic meter. Given that the volume of the room is 100 cubic meters andit initially contains no carbon monoxide, how long will it take for the alarm to sound?