Question 1

In the previous problem, how much salt will there be in the tank after a long period of time?

Accepted Answer

A)  1000 kilograms 
B)  300 kilograms 
C)  120 kilograms 
D)  80 kilograms 
E)  none of the above 
A)  1000 kilograms 
B)  300 kilograms 
C)  120 kilograms 
D)  80 kilograms 
E)  none of the above

Question 2

In the previous problem, how much salt will there be in tanks A and B after a long period of time?

Accepted Answer

A)  5 pounds in A, 20 pounds in B 
B)  20 pounds in A, 5 pounds in B 
C)  5 pounds in A, 40 pounds in B 
D)  40 pounds in A, 15 pounds in B 
E)  none of the above 
A)  5 pounds in A, 20 pounds in B 
B)  20 pounds in A, 5 pounds in B 
C)  5 pounds in A, 40 pounds in B 
D)  40 pounds in A, 15 pounds in B 
E)  none of the above

Question 3

In Newton's Law of cooling, $\frac { d T } { d t } = k \left( T - T _ { m } \right)$ the constant k is

Accepted Answer

A)  a constant of integration evaluated from an initial condition 
B)  a constant of integration evaluated from another condition 
C)  a proportionality constant evaluated from an initial condition 
D)  a proportionality constant evaluated from another condition 
A)  a constant of integration evaluated from an initial condition 
B)  a constant of integration evaluated from another condition 
C)  a proportionality constant evaluated from an initial condition 
D)  a proportionality constant evaluated from another condition

Question 4

The differential equation $\frac { d P } { d t } = ( k \cos t ) P$ , where k is a positive constant, models a population that undergoes yearly fluctuations. The solution of the equation is

Accepted Answer

A)  $P = e ^ { c k \sin t }$ 
B)  $P = c e ^ { k \cos t }$ 
C)  $P = c e ^ { - k \cos t }$ 
D)  $P = c e ^ { - k \sin t }$ 
E)  $P = c e ^ { k \sin t }$ 
A)  $P = e ^ { c k \sin t }$ 
B)  $P = c e ^ { k \cos t }$ 
C)  $P = c e ^ { - k \cos t }$ 
D)  $P = c e ^ { - k \sin t }$ 
E)  $P = c e ^ { k \sin t }$

Question 5

An object is taken out of a $65 ^ { \circ } \mathrm { F }$ room and placed outside where the temperature is $35 ^ { \circ } \mathrm { F }$ room. Five minutes later the temperature is $63 ^ { \circ } \mathrm { F }$ . It cools according to Newton's Law. The temperature of the object after one hour is

Accepted Answer

A)  $50.5 ^ { \circ } \mathrm { F }$ 
B)  $49.9 ^ { \circ } \mathrm { F }$ 
C)  $49.3 ^ { \circ } \mathrm { F }$ 
D)  $48.7 ^ { \circ } \mathrm { F }$ 
E)  $48.1 ^ { \circ } \mathrm { F }$ 
A)  $50.5 ^ { \circ } \mathrm { F }$ 
B)  $49.9 ^ { \circ } \mathrm { F }$ 
C)  $49.3 ^ { \circ } \mathrm { F }$ 
D)  $48.7 ^ { \circ } \mathrm { F }$ 
E)  $48.1 ^ { \circ } \mathrm { F }$

Question 6

A chicken is taken out of the freezer $\left( 0 ^ { \circ } \mathrm { C } \right)$ and placed on a table in a $23 ^ { \circ } \mathrm { C }$ room. Forty-five minutes later the temperature is $10 ^ { \circ } \mathrm { C }$ . It warms according to Newton's Law. How long does it take before the temperature reaches $20 ^ { \circ } \mathrm { C }$ ?

Accepted Answer

A)  147 minutes 
B)  153 minutes 
C)  157 minutes 
D)  161 minutes 
E)  165 minutes 
A)  147 minutes 
B)  153 minutes 
C)  157 minutes 
D)  161 minutes 
E)  165 minutes

Question 7

In the previous two problems, the amount of salt in the tank at time t is

Accepted Answer

A)  $A ( t ) = - 75 + 77 e ^ { 2 t / 25 }$ 
B)  $A ( t ) = 75 - 73 e ^ { - 2 t / 25 }$ 
C)  $A ( t ) = 50 - 48 e ^ { - 2 t / 25 }$ 
D)  $A ( t ) = - 50 + 52 e ^ { 2 t / 25 }$ 
E)  $A ( t ) = 75 - 73 e ^ { 2 t / 25 }$ 
A)  $A ( t ) = - 75 + 77 e ^ { 2 t / 25 }$ 
B)  $A ( t ) = 75 - 73 e ^ { - 2 t / 25 }$ 
C)  $A ( t ) = 50 - 48 e ^ { - 2 t / 25 }$ 
D)  $A ( t ) = - 50 + 52 e ^ { 2 t / 25 }$ 
E)  $A ( t ) = 75 - 73 e ^ { 2 t / 25 }$

Question 8

In the previous problem, how much salt will there be in tanks A and B after a long period of time?

Accepted Answer

A)  3 pounds in A, 2 pounds in B 
B)  40 pounds in A, 24 pounds in B 
C)  0 pounds in A, 0 pounds in B 
D)  40 pounds in A, 30 pounds in B 
E)  none of the above 
A)  3 pounds in A, 2 pounds in B 
B)  40 pounds in A, 24 pounds in B 
C)  0 pounds in A, 0 pounds in B 
D)  40 pounds in A, 30 pounds in B 
E)  none of the above

Question 9

The half-life of radium is 1700 years. Assume that the decay rate is proportional to the amount. An initial amount of 5 grams of radium deca to 3 grams in

Accepted Answer

A)  850 years 
B)  1050 years 
C)  1150 years 
D)  1250 years 
E)  1350 years 
A)  850 years 
B)  1050 years 
C)  1150 years 
D)  1250 years 
E)  1350 years

Question 10

A tank contains 50 gallons of water in which 2 pounds of salt is dissolved. A brine solution containing 1.5 pounds of salt per gallon of water is pumped into the tank at the rate of 4 gallons per minute, and the well-stirred mixture is pumped out at the same rate. Let $A ( t )$ represent the amount of salt in the tank at time t. The correct initial value problem for $A ( t )$ is

Accepted Answer

A)  $\frac { d A } { d t } = 6 - 2 A / 25 , A ( 0 ) = 2$ 
B)  $\frac { d A } { d t } = 6 + 2 A / 25 , A ( 0 ) = 2$ 
C)  $\frac { d A } { d t } = 4 + 2 A / 25 , A ( 0 ) = 2$ 
D)  $\frac { d A } { d t } = 1.5 - 2 A / 25 , A ( 0 ) = 0$ 
E)  $\frac { d A } { d t } = 4 - 2 A / 25 , A ( 0 ) = 0$ 
A)  $\frac { d A } { d t } = 6 - 2 A / 25 , A ( 0 ) = 2$ 
B)  $\frac { d A } { d t } = 6 + 2 A / 25 , A ( 0 ) = 2$ 
C)  $\frac { d A } { d t } = 4 + 2 A / 25 , A ( 0 ) = 2$ 
D)  $\frac { d A } { d t } = 1.5 - 2 A / 25 , A ( 0 ) = 0$ 
E)  $\frac { d A } { d t } = 4 - 2 A / 25 , A ( 0 ) = 0$

Question 11

In Newton's Law of cooling, $\frac { d T } { d t } = k \left( T - T _ { m } \right) , T _ { m }$ is

Accepted Answer

A)  the temperature of the object 
B)  the temperature of the environment 
C)  the initial temperature 
D)  the temperature after a specified period of time 
E)  none of the above 
A)  the temperature of the object 
B)  the temperature of the environment 
C)  the initial temperature 
D)  the temperature after a specified period of time 
E)  none of the above

Question 12

The solution of the system of differential equations in the previous problem is

Accepted Answer

A)  $x = c _ { 1 } e ^ { - 0.3 t } , y = - 3 c _ { 1 } e ^ { - 03 t } + c _ { 2 } e ^ { - 02 t } , z = c _ { 3 } - 3 c _ { 1 } e ^ { - 0.3 t } - c _ { 2 } e ^ { - 0.2 t }$. 
B)  $x = c _ { 1 } e ^ { - 02 t } , y = - 2 c _ { 1 } e ^ { - 0.2 t } + c _ { 2 } e ^ { - 0.3 t } , z = c _ { 3 } + 2 c _ { 1 } e ^ { - 0.3 t } - c _ { 2 } e ^ { - 0.2 t }$ . 
C)  $x = c _ { 1 } e ^ { - 0.3 t } , y = - 3 c _ { 1 } e ^ { - 03 t } + c _ { 2 } e ^ { - 02 t } , z = c _ { 3 } + 2 c _ { 1 } e ^ { - 0.3 t } + c _ { 2 } e ^ { - 0.2 t }$ 
D)  $x = c _ { 1 } e ^ { - 0.3 t } , y = - 3 c _ { 1 } e ^ { - 03 t } + c _ { 2 } e ^ { - 02 t } , z = c _ { 3 } + 2 c _ { 1 } e ^ { - 0.3 t } - c _ { 2 } e ^ { - 0.2 t }$. 
E)  $x = c _ { 1 } e ^ { - 02 t } , y = - 2 c _ { 1 } e ^ { - 0.2 t } + c _ { 2 } e ^ { - 0.3 t } , z = c _ { 3 } + 2 c _ { 1 } e ^ { - 0.3 t } - c _ { 2 } e ^ { - 0.2 t }$ 
A)  $x = c _ { 1 } e ^ { - 0.3 t } , y = - 3 c _ { 1 } e ^ { - 03 t } + c _ { 2 } e ^ { - 02 t } , z = c _ { 3 } - 3 c _ { 1 } e ^ { - 0.3 t } - c _ { 2 } e ^ { - 0.2 t }$. 
B)  $x = c _ { 1 } e ^ { - 02 t } , y = - 2 c _ { 1 } e ^ { - 0.2 t } + c _ { 2 } e ^ { - 0.3 t } , z = c _ { 3 } + 2 c _ { 1 } e ^ { - 0.3 t } - c _ { 2 } e ^ { - 0.2 t }$ . 
C)  $x = c _ { 1 } e ^ { - 0.3 t } , y = - 3 c _ { 1 } e ^ { - 03 t } + c _ { 2 } e ^ { - 02 t } , z = c _ { 3 } + 2 c _ { 1 } e ^ { - 0.3 t } + c _ { 2 } e ^ { - 0.2 t }$ 
D)  $x = c _ { 1 } e ^ { - 0.3 t } , y = - 3 c _ { 1 } e ^ { - 03 t } + c _ { 2 } e ^ { - 02 t } , z = c _ { 3 } + 2 c _ { 1 } e ^ { - 0.3 t } - c _ { 2 } e ^ { - 0.2 t }$. 
E)  $x = c _ { 1 } e ^ { - 02 t } , y = - 2 c _ { 1 } e ^ { - 0.2 t } + c _ { 2 } e ^ { - 0.3 t } , z = c _ { 3 } + 2 c _ { 1 } e ^ { - 0.3 t } - c _ { 2 } e ^ { - 0.2 t }$

Question 13

The amount of salt in the tank at time t in the previous two problems is

Accepted Answer

A)  $A ( t ) = - 200 + 200.3 e ^ { t / 40 }$ 
B)  $A ( t ) = 200 - 199.7 e ^ { - t / 40 }$ 
C)  $A ( t ) = 80 - 79.7 e ^ { - t / 40 }$ 
D)  $A ( t ) = - 80 + 80.3 e ^ { t / 25 }$ 
E)  $A ( t ) = 200 + 100 e ^ { - t / 40 }$ 
A)  $A ( t ) = - 200 + 200.3 e ^ { t / 40 }$ 
B)  $A ( t ) = 200 - 199.7 e ^ { - t / 40 }$ 
C)  $A ( t ) = 80 - 79.7 e ^ { - t / 40 }$ 
D)  $A ( t ) = - 80 + 80.3 e ^ { t / 25 }$ 
E)  $A ( t ) = 200 + 100 e ^ { - t / 40 }$

Question 14

A ball is thrown upward from the top of a 200 foot tall building with a velocity of 40 feet per second. Take the positive direction upward and the origin of the coordinate system at ground level. What is the initial value problem for the position, $x ( t )$ , of the ball at time t?

Accepted Answer

A)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } = 40 , x ( 0 ) = 200 , \frac { d x } { d t } ( 0 ) = 40$ 
B)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } = - 40 , x ( 0 ) = 200 , \frac { d x } { d t } ( 0 ) = 40$ 
C)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } = 32 , x ( 0 ) = 200 , \frac { d x } { d t } ( 0 ) = 40$ 
D)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } = - 32 , x ( 0 ) = 200 , \frac { x A } { d t } ( 0 ) = 40$ 
E)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } = 200 , x ( 0 ) = 32 , \frac { d x } { d t } ( 0 ) = 40$ 
A)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } = 40 , x ( 0 ) = 200 , \frac { d x } { d t } ( 0 ) = 40$ 
B)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } = - 40 , x ( 0 ) = 200 , \frac { d x } { d t } ( 0 ) = 40$ 
C)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } = 32 , x ( 0 ) = 200 , \frac { d x } { d t } ( 0 ) = 40$ 
D)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } = - 32 , x ( 0 ) = 200 , \frac { x A } { d t } ( 0 ) = 40$ 
E)  $\frac { d ^ { 2 } x } { d t ^ { 2 } } = 200 , x ( 0 ) = 32 , \frac { d x } { d t } ( 0 ) = 40$

Question 15

The solution of the system of differential equations in the two previous problems is

Accepted Answer

A)  $x = 10 e ^ { - 0.1 t } , y = 12.5 \left( e ^ { - 0.1 t } - e ^ { - 0.5 t } \right) , z = 10 - 12.5 e ^ { - 0.1 t } + 2.5 e ^ { - 0.5 t }$ 
B)  $x = 10 e ^ { - 0.1 t } , y = 12.5 \left( e ^ { - 0.5 t } - e ^ { - 0.1 t } \right) , z = 10 - 12.5 e ^ { - 0.5 t } + 2.5 e ^ { - 0.1 t }$ 
C)  $x = 10 e ^ { - 0.5 t } , y = 12.5 \left( e ^ { - 0.5 t } - e ^ { - 0.1 t } \right) , z = 10 - 12.5 e ^ { - 0.2 t } + 2.5 e ^ { - 0.3 t }$ 
D)  $x = 10 e ^ { - 0.5 t } , y = 12.5 \left( e ^ { - 0.1 t } - e ^ { - 0.5 t } \right) , z = 10 - 12.5 e ^ { - 0.5 t } + 2.5 e ^ { - 0.1 t }$ 
E)  $x = 10 e ^ { - 0.5 t } , y = 12.5 \left( e ^ { - 0.1 t } - e ^ { - 0.5 t } \right) , z = 10 - 12.5 e ^ { - 0.1 t } + 2.5 e ^ { - 0.5 t }$ 
A)  $x = 10 e ^ { - 0.1 t } , y = 12.5 \left( e ^ { - 0.1 t } - e ^ { - 0.5 t } \right) , z = 10 - 12.5 e ^ { - 0.1 t } + 2.5 e ^ { - 0.5 t }$ 
B)  $x = 10 e ^ { - 0.1 t } , y = 12.5 \left( e ^ { - 0.5 t } - e ^ { - 0.1 t } \right) , z = 10 - 12.5 e ^ { - 0.5 t } + 2.5 e ^ { - 0.1 t }$ 
C)  $x = 10 e ^ { - 0.5 t } , y = 12.5 \left( e ^ { - 0.5 t } - e ^ { - 0.1 t } \right) , z = 10 - 12.5 e ^ { - 0.2 t } + 2.5 e ^ { - 0.3 t }$ 
D)  $x = 10 e ^ { - 0.5 t } , y = 12.5 \left( e ^ { - 0.1 t } - e ^ { - 0.5 t } \right) , z = 10 - 12.5 e ^ { - 0.5 t } + 2.5 e ^ { - 0.1 t }$ 
E)  $x = 10 e ^ { - 0.5 t } , y = 12.5 \left( e ^ { - 0.1 t } - e ^ { - 0.5 t } \right) , z = 10 - 12.5 e ^ { - 0.1 t } + 2.5 e ^ { - 0.5 t }$

Question 16

In the competition model $\frac { d x } { d t } = - a x + b x y , \frac { d y } { d t } = e y - c x y$ where $x ( t )$ and $y ( t )$ are the populations of the competing species, moose and deer, respectively, the coefficient c represents which of the following:

Accepted Answer

A)  the moose growth rate 
B)  the deer growth rate 
C)  the decrease in the moose population due to interactions with the deer 
D)  the decrease in the deer population due to interactions with the moose 
E)  none of the above 
A)  the moose growth rate 
B)  the deer growth rate 
C)  the decrease in the moose population due to interactions with the deer 
D)  the decrease in the deer population due to interactions with the moose 
E)  none of the above

Question 17

A chicken is taken out of the freezer $\left( 0 ^ { \circ } \mathrm { C } \right)$ and placed on a table in a $20 ^ { \circ } \mathrm { C }$ room. Ten minutes later the temperature is $2 ^ { \circ } \mathrm { C }$ . It warms according to Newton's Law. How long does it take before the temperature reaches $15 ^ { \circ } \mathrm { C }$ ?

Accepted Answer

A)  122 minutes 
B)  127 minutes 
C)  132 minutes 
D)  137 minutes 
E)  142 minutes 
A)  122 minutes 
B)  127 minutes 
C)  132 minutes 
D)  137 minutes 
E)  142 minutes

Question 18

Two chemicals, A and B, are combined, forming chemical C. The rate of the reaction is jointly proportional to the amounts of A and B not yet converted to C. Initially, there are 200 grams of A and 300 grams of B, and, during the reaction, for each gram of A used up in the conversion, there are three grams of B used up. An experiments shows that 75 grams of C are produced in the first ten minutes. After a long period of time, how much of A and of B remains, and how much of C has been produced?

Accepted Answer

A)  200 grams of A, 0 grams of B, 300 grams of C 
B)  0 grams of A, 0 grams of B, 500 grams of C 
C)  100 grams of A, 0 grams of B, 400 grams of C 
D)  0 grams of A, 100 grams of B, 400 grams of C 
E)  0 grams of A, 200 grams of B, 300 grams of C 
A)  200 grams of A, 0 grams of B, 300 grams of C 
B)  0 grams of A, 0 grams of B, 500 grams of C 
C)  100 grams of A, 0 grams of B, 400 grams of C 
D)  0 grams of A, 100 grams of B, 400 grams of C 
E)  0 grams of A, 200 grams of B, 300 grams of C

Question 19

Tank A contains 50 gallons of water in which 2 pounds of salt has been dissolved. Tank B contains 30 gallons of water in which 3 pounds of salt has been dissolved. A brine mixture with a concentration of 0.8 pounds of salt per gallon of water is pumped into tank A at the rate of 3 gallons per minute. The well-mixed solution is then pumped from tank A to tank B at the rate of 4 gallons per minute. The solution from tank B is also pumped through another pipe into tank A at the rate of 1 gallonper minute, and the solution from tank B is also pumped out of the system at the rate of 3 gallons per minute. The correct differential equations with initial conditions for the amounts, $x ( t )$ and $y ( t )$ , of salt in tanks A and B, respectively, at time t are

Accepted Answer

A)  $\frac { d x } { d t } = 3 - 2 x / 25 + y / 5 , \frac { d y } { d t } = x / 25 - y / 15 , x ( 0 ) = 2 , y ( 0 ) = 3$ 
B)  $\frac { d x } { d t } = 3 - x / 25 + y / 15 , \frac { d y } { d t } = 2 x / 25 - 2 y / 15 , x ( 0 ) = 2 , y ( 0 ) = 3$ 
C)  $\frac { d x } { d t } = 2.4 - 2 x / 25 + y / 30 , \frac { d y } { d t } = 2 x / 25 - 2 y / 15 , x ( 0 ) = 2 , y ( 0 ) = 3$ 
D)  $\frac { d x } { d t } = 2.4 - x / 50 + y / 30 , \frac { d y } { d t } = x / 40 - y / 3 , x ( 0 ) = 2 , y ( 0 ) = 3$ 
E)  $\frac { d x } { d t } = 2.4 - x / 25 + y / 15 , \frac { d y } { d t } = x / 50 - y / 30 , x ( 0 ) = 2 , y ( 0 ) = 3$ 
A)  $\frac { d x } { d t } = 3 - 2 x / 25 + y / 5 , \frac { d y } { d t } = x / 25 - y / 15 , x ( 0 ) = 2 , y ( 0 ) = 3$ 
B)  $\frac { d x } { d t } = 3 - x / 25 + y / 15 , \frac { d y } { d t } = 2 x / 25 - 2 y / 15 , x ( 0 ) = 2 , y ( 0 ) = 3$ 
C)  $\frac { d x } { d t } = 2.4 - 2 x / 25 + y / 30 , \frac { d y } { d t } = 2 x / 25 - 2 y / 15 , x ( 0 ) = 2 , y ( 0 ) = 3$ 
D)  $\frac { d x } { d t } = 2.4 - x / 50 + y / 30 , \frac { d y } { d t } = x / 40 - y / 3 , x ( 0 ) = 2 , y ( 0 ) = 3$ 
E)  $\frac { d x } { d t } = 2.4 - x / 25 + y / 15 , \frac { d y } { d t } = x / 50 - y / 30 , x ( 0 ) = 2 , y ( 0 ) = 3$

Question 20

Radioactive element X decays to element Y with decay constant -0.3. Y decays to stable element Z with decay constant $- 0.2$ . What is the system of differential equations for the amounts, $x ( t ) , y ( t ) , z ( t )$ of the elements X, Y, Z, respectively, at time t.

Accepted Answer

A)  $\frac { d x } { d t } = - 0.2 x , \frac { d y } { d t } = 0.2 x - 0.3 y , \frac { d z } { d t } = 0.2 y$ 
B)  $\frac { d x } { d t } = - 0.2 x , \frac { d y } { d t } = 0.2 x - 0.3 y , \frac { d z } { d t } = 0.3 y$ 
C)  $\frac { d x } { d t } = - 0.3 x , \frac { d y } { d t } = 0.3 x - 0.2 y , \frac { d z } { d t } = 0.2 y$ 
D)  $\frac { d x } { d t } = - 0.3 x , \frac { d y } { d t } = 0.3 x - 0.2 y , \frac { d z } { d t } = 0.3 y$ 
E)  $\frac { d x } { d t } = - 0.3 y , \frac { d y } { d t } = 0.3 x - 0.2 y , \frac { d z } { d t } = 0.2 y$ 
A)  $\frac { d x } { d t } = - 0.2 x , \frac { d y } { d t } = 0.2 x - 0.3 y , \frac { d z } { d t } = 0.2 y$ 
B)  $\frac { d x } { d t } = - 0.2 x , \frac { d y } { d t } = 0.2 x - 0.3 y , \frac { d z } { d t } = 0.3 y$ 
C)  $\frac { d x } { d t } = - 0.3 x , \frac { d y } { d t } = 0.3 x - 0.2 y , \frac { d z } { d t } = 0.2 y$ 
D)  $\frac { d x } { d t } = - 0.3 x , \frac { d y } { d t } = 0.3 x - 0.2 y , \frac { d z } { d t } = 0.3 y$ 
E)  $\frac { d x } { d t } = - 0.3 y , \frac { d y } { d t } = 0.3 x - 0.2 y , \frac { d z } { d t } = 0.2 y$

In the previous problem, how much salt will there be in the tank after a long period of time?

In the previous problem, how much salt will there be in tanks A and B after a long period of time?

In Newton's Law of cooling, $\frac { d T } { d t } = k \left( T - T _ { m } \right)$ the constant k is

The differential equation $\frac { d P } { d t } = ( k \cos t ) P$ , where k is a positive constant, models a population that undergoes yearly fluctuations. The solution of the equation is

An object is taken out of a $65 ^ { \circ } \mathrm { F }$ room and placed outside where the temperature is $35 ^ { \circ } \mathrm { F }$ room. Five minutes later the temperature is $63 ^ { \circ } \mathrm { F }$ . It cools according to Newton's Law. The temperature of the object after one hour is

In the previous two problems, the amount of salt in the tank at time t is

In the previous problem, how much salt will there be in tanks A and B after a long period of time?

The half-life of radium is 1700 years. Assume that the decay rate is proportional to the amount. An initial amount of 5 grams of radium deca to 3 grams in

In Newton's Law of cooling, $\frac { d T } { d t } = k \left( T - T _ { m } \right) , T _ { m }$ is

The solution of the system of differential equations in the previous problem is

The amount of salt in the tank at time t in the previous two problems is

A ball is thrown upward from the top of a 200 foot tall building with a velocity of 40 feet per second. Take the positive direction upward and the origin of the coordinate system at ground level. What is the initial value problem for the position, $x ( t )$ , of the ball at time t?

The solution of the system of differential equations in the two previous problems is

In the competition model $\frac { d x } { d t } = - a x + b x y , \frac { d y } { d t } = e y - c x y$ where $x ( t )$ and $y ( t )$ are the populations of the competing species, moose and deer, respectively, the coefficient c represents which of the following:

Radioactive element X decays to element Y with decay constant -0.3. Y decays to stable element Z with decay constant $- 0.2$ . What is the system of differential equations for the amounts, $x ( t ) , y ( t ) , z ( t )$ of the elements X, Y, Z, respectively, at time t.

Introduction to Differential Equations

First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

Filters

Exam 3: Modeling With First-Order Differential Equations

In the previous problem, how much salt will there be in the tank after a long period of time?

In the previous problem, how much salt will there be in tanks A and B after a long period of time?

In Newton's Law of cooling, dTdt=k(T−Tm)\frac { d T } { d t } = k \left( T - T _ { m } \right)dtdT​=k(T−Tm​) the constant k is

The differential equation dPdt=(kcos⁡t)P\frac { d P } { d t } = ( k \cos t ) PdtdP​=(kcost)P , where k is a positive constant, models a population that undergoes yearly fluctuations. The solution of the equation is

In the previous two problems, the amount of salt in the tank at time t is

In the previous problem, how much salt will there be in tanks A and B after a long period of time?

The half-life of radium is 1700 years. Assume that the decay rate is proportional to the amount. An initial amount of 5 grams of radium deca to 3 grams in

In Newton's Law of cooling, dTdt=k(T−Tm),Tm\frac { d T } { d t } = k \left( T - T _ { m } \right) , T _ { m }dtdT​=k(T−Tm​),Tm​ is

The solution of the system of differential equations in the previous problem is

The amount of salt in the tank at time t in the previous two problems is

A ball is thrown upward from the top of a 200 foot tall building with a velocity of 40 feet per second. Take the positive direction upward and the origin of the coordinate system at ground level. What is the initial value problem for the position, x(t)x ( t )x(t) , of the ball at time t?

The solution of the system of differential equations in the two previous problems is

Introduction to Differential Equations

First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

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In Newton's Law of cooling, $\frac { d T } { d t } = k \left( T - T _ { m } \right)$ the constant k is

The differential equation $\frac { d P } { d t } = ( k \cos t ) P$ , where k is a positive constant, models a population that undergoes yearly fluctuations. The solution of the equation is

In Newton's Law of cooling, $\frac { d T } { d t } = k \left( T - T _ { m } \right) , T _ { m }$ is

A ball is thrown upward from the top of a 200 foot tall building with a velocity of 40 feet per second. Take the positive direction upward and the origin of the coordinate system at ground level. What is the initial value problem for the position, $x ( t )$ , of the ball at time t?