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Deck 6: Differential Equations
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Question
A population of rabbits in a certain habitat grows according to the differential equation 
where t is measured in months 
and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is, 
.
Estimates of y(t)can be produced using Euler's Method with step size
.To the nearest rabbit,the estimate for y(2)is
A)176
B)358
C)353
D)349
where t is measured in months and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is, . Estimates of y(t)can be produced using Euler's Method with step size
.To the nearest rabbit,the estimate for y(2)is
A)176
B)358
C)353
D)349
Question
Water flows continuously from a large tank at a rate proportional to the amount of water remaining in the tank;that is, 
.There was initially 10,000 cubic feet of water in the tank,and at time 
hours,6,000 cubic feet of water remained.
To the nearest cubic foot,how much water remained in the tank at time
hours?
A)4,937
B)5,170
C)5,394
D)4,891
.There was initially 10,000 cubic feet of water in the tank,and at time hours,6,000 cubic feet of water remained. To the nearest cubic foot,how much water remained in the tank at time
hours?
A)4,937
B)5,170
C)5,394
D)4,891
Question
Consider the differential equation 
,for 
only,with initial value 
.
Which of the following is the slope field for the general solution to the given differential equation?
A)
B)
C)
D)
,for only,with initial value .
Which of the following is the slope field for the general solution to the given differential equation?
A)
B)
C)
D)
Question
A population of rabbits in a certain habitat grows according to the differential equation 
where t is measured in months 
and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is, 
.
What is the fastest growth rate,in rabbits per month,that this population exhibits?
A)150
B)600
C)300
D)250
where t is measured in months and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is, . What is the fastest growth rate,in rabbits per month,that this population exhibits?
A)150
B)600
C)300
D)250
Question
Consider the differential equation 
,for which the solution is g(x). Let y(x)be the particular solution that contains (0,1).Using Euler's Method with step size 
,what is the estimate for 
?
A)4
B)3.5
C)3
D)4.5
,for which the solution is g(x). Let y(x)be the particular solution that contains (0,1).Using Euler's Method with step size ,what is the estimate for ?
A)4
B)3.5
C)3
D)4.5
Question
Each of the following graphs is from a logistic function 
.Which one has the smallest value of b?
A)
B)
C)
D)
E)
.Which one has the smallest value of b? A)
B)
C)
D)
E)
Question
Water is being pumped continuously into a tank at a rate that is inversely proportional to the amount of water in the tank;that is, 
,where y is the number of gallons of water in the tank after t minutes 
.Initially there were 5 gallons of water in the tank,and after 3 inutes there were 8 gallons.How many gallons of water were in the tank at time 20 minutes?
A)
B)
C)
D)
,where y is the number of gallons of water in the tank after t minutes .Initially there were 5 gallons of water in the tank,and after 3 inutes there were 8 gallons.How many gallons of water were in the tank at time 20 minutes? A)
B)
C)
D)
Question
Consider the differential equation 
,for which the solution is g(x). Which of the following statements is true about the particular solution that contains (1,-4)?
A)g(x)is increasing and concave up.
B)g(x)is increasing and concave down.
C)g(x)is decreasing and concave up.
D)g(x)is decreasing and concave down.
,for which the solution is g(x). Which of the following statements is true about the particular solution that contains (1,-4)?
A)g(x)is increasing and concave up.
B)g(x)is increasing and concave down.
C)g(x)is decreasing and concave up.
D)g(x)is decreasing and concave down.
Question
The general solution to the differential equation 
is
A)
B)
C)
D)
is A)
B)
C)
D)
Question
If 
then the particular solution 
is
A)
B)
C)
D)
then the particular solution is A)
B)
C)
D)
Question
For what value(s)of x is 
?
? Question
Solve the differential equation in 
to find f(x)in terms of x only.
to find f(x)in terms of x only. Question
Consider the differential equation 
,for 
only,with initial value 
.
The particular solution y(x)is
A)
B)
C)
D)
,for only,with initial value .
The particular solution y(x)is
A)
B)
C)
D)
Question
A population of rabbits in a certain habitat grows according to the differential equation 
where t is measured in months 
and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is, 
.
Which of the following slope fields represents an approximate general solution to the given differential equation?
A)
B)
C)
D)
where t is measured in months and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is, . Which of the following slope fields represents an approximate general solution to the given differential equation?
A)
B)
C)
D)
Question
Identify the graph of the logistic function 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Consider the differential equation 
,for which the solution is 
.Let 
. The particular solution is
A)
B)
C)
D)
,for which the solution is .Let . The particular solution is
A)
B)
C)
D)
Question
A differentiable function
is defined such that,for all values of 
in its domain, 
.
What is the domain of
?
A differentiable function
is defined such that,for all values of in its domain, .What is the domain of
? Question
Water flows continuously from a large tank at a rate proportional to the amount of water remaining in the tank;that is, 
.There was initially 9,000 cubic feet of water in the tank,and at time 
hours,8,000 cubic feet of water remained.
What is the value of k in the equation
? Round your answer to 3 decimal places.
A)0.014
B)0.060
C)-0.131
D)-0.059
.There was initially 9,000 cubic feet of water in the tank,and at time hours,8,000 cubic feet of water remained. What is the value of k in the equation
? Round your answer to 3 decimal places.
A)0.014
B)0.060
C)-0.131
D)-0.059
Question
Show that 
.
. Question
Consider the differential equation 
,for 
only,with initial value 
.
Using Euler's Method with step size
,what is the estimate for 
?
A)
B)2.5
C)3
D)4.85
,for only,with initial value .
Using Euler's Method with step size
,what is the estimate for ?
A)
B)2.5
C)3
D)4.85
Question
The logistic function 
models the growth of a population.Identify the initial population.
A)10
B)12
C)5
D)60
E)6
models the growth of a population.Identify the initial population. A)10
B)12
C)5
D)60
E)6
Question
Match the logistic differential equation and initial condition with the graph of its solution shown below. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the particular solution of the differential equation 
that satisfies the initial condition 
.
A)
B)
C)
D)
E)
that satisfies the initial condition . A)
B)
C)
D)
E)
Question
Each of the following graphs is from a logistic function 
.Which one has the largest value of b?
A)
B)
C)
D)
E)
.Which one has the largest value of b? A)
B)
C)
D)
E)
Question
The logistic function 
models the growth of a population.Determine when the population reaches 40% of the maximum carrying capacity.Round your answer to three decimal places.
A)4.317
B)3.000
C)0.474
D)0.677
E)0.301
models the growth of a population.Determine when the population reaches 40% of the maximum carrying capacity.Round your answer to three decimal places. A)4.317
B)3.000
C)0.474
D)0.677
E)0.301
Question
A conservation organization releases 30 panthers into a preserve.After 3 years,there are 50 panthers in the preserve.The preserve has a carrying capacity of 150.Write a logistic function that models the population of panthers in the preserve.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Determine whether the function 
is homogeneous and determine its degree if it is.
A)homogeneous,the degree is 3
B)homogeneous,the degree is 4
C)not homogeneous00
D)homogeneous,the degree is 2
E)homogeneous,the degree is 1
is homogeneous and determine its degree if it is. A)homogeneous,the degree is 3
B)homogeneous,the degree is 4
C)not homogeneous00
D)homogeneous,the degree is 2
E)homogeneous,the degree is 1
Question
Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Match the logistic equation and initial condition with the graph of the solution. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Question
Find the logistic equation that satisfies the following differential equation and initial condition. 
A)
B)
C)
D)
E)none of the above
A)
B)
C)
D)
E)none of the above
Question
The logistic function 
models the growth of a population.Determine when the population reaches one-half of the maximum carrying capacity.Round your answer to three decimal places.
A)0.549
B)3.333
C)1.151
D)5.000
E)1.000
models the growth of a population.Determine when the population reaches one-half of the maximum carrying capacity.Round your answer to three decimal places. A)0.549
B)3.333
C)1.151
D)5.000
E)1.000
Question
Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
A calf that weighs 75 pounds at birth gains weight at the rate 
where w is weight in pounds and t is time in years.Use a computer algebra system to solve the differential equation for 
.
A)
B)
C)
D)
E)
where w is weight in pounds and t is time in years.Use a computer algebra system to solve the differential equation for . A)
B)
C)
D)
E)
Question
The logistic function 
models the growth of a population.Identify the value of k.
A)3.5
B)4
C)2
D)10
E)3
models the growth of a population.Identify the value of k. A)3.5
B)4
C)2
D)10
E)3
Question
Find the general solution of the differential equation 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Find an equation of the graph that passes through the point (2,5)and has the slope 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Question
Solve the homogeneous differential equation 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
The logistic function 
models the growth of a population.Identify the maximum carrying capacity.
A)15
B)5.5
C)5
D)4
E)6
models the growth of a population.Identify the maximum carrying capacity. A)15
B)5.5
C)5
D)4
E)6
Question
Write and solve the differential equation that models the following verbal statement:
The rate of change of M with respect to s is proportional to
.
A)
B)
C)
D)
E)
The rate of change of M with respect to s is proportional to
.
A)
B)
C)
D)
E)
Question
The isotope 
has a half-life of 24,100 years.After 1,000 years,a sample of the isotope is reduced to 1.4 grams.What was the initial size of the sample (in grams)? How much will remain after 10,000 years (i.e. ,after another 9,000 years)? Round your answers to four decimal places.
A)1.0086 g,0.7565 g
B)2.3054 g,1.7291 g
C)1.4409 g,1.0807 g
D)2.0172 g,1.5130 g
E)1.8731 g,1.4049 g
has a half-life of 24,100 years.After 1,000 years,a sample of the isotope is reduced to 1.4 grams.What was the initial size of the sample (in grams)? How much will remain after 10,000 years (i.e. ,after another 9,000 years)? Round your answers to four decimal places. A)1.0086 g,0.7565 g
B)2.3054 g,1.7291 g
C)1.4409 g,1.0807 g
D)2.0172 g,1.5130 g
E)1.8731 g,1.4049 g
Question
Solve the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
The isotope 
has a half-life of 5,715 years.After 20,000 years,a sample of the isotope is reduced 0.7 grams.What was the initial size of the sample (in grams)? How large was the sample after the first 2,000 years? Round your answers to four decimal places.
A)7.9172 g,6.2119 g
B)10.2924 g,8.0755 g
C)4.7503 g,3.7271 g
D)6.3338 g,4.9695 g
E)3.9586 g,3.1059 g
has a half-life of 5,715 years.After 20,000 years,a sample of the isotope is reduced 0.7 grams.What was the initial size of the sample (in grams)? How large was the sample after the first 2,000 years? Round your answers to four decimal places. A)7.9172 g,6.2119 g
B)10.2924 g,8.0755 g
C)4.7503 g,3.7271 g
D)6.3338 g,4.9695 g
E)3.9586 g,3.1059 g
Question
Solve the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Solve the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Solve the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
A calf that weighs 60 pounds at birth gains weight at the rate 
where w is weight in pounds and t is time in years.If the animal is sold when its weight reaches 650 pounds,find the time of sale using the model 
.Round your answer to two decimal places.
A)7.31 years
B)2.65 years
C)1.07 years
D)0.72 year
E)0.29 year
where w is weight in pounds and t is time in years.If the animal is sold when its weight reaches 650 pounds,find the time of sale using the model .Round your answer to two decimal places. A)7.31 years
B)2.65 years
C)1.07 years
D)0.72 year
E)0.29 year
Question
Question
The isotope 
has a half-life of 24,100 years.Given an initial amount of 6 grams of the isotope,how many grams will remain after 500 years? After 5,000 years? Round your answers to four decimal places.
A)4.1400 g,3.6374 g
B)3.5486 g,3.1178 g
C)5.9143 g,5.1963 g
D)2.3657 g,2.0785 g
E)7.0972 g,6.2356 g
has a half-life of 24,100 years.Given an initial amount of 6 grams of the isotope,how many grams will remain after 500 years? After 5,000 years? Round your answers to four decimal places. A)4.1400 g,3.6374 g
B)3.5486 g,3.1178 g
C)5.9143 g,5.1963 g
D)2.3657 g,2.0785 g
E)7.0972 g,6.2356 g
Question
Find the exponential function 
that passes through the two given points.Round your values of C and k to four decimal places. 
A)
B)
C)
D)
E)
that passes through the two given points.Round your values of C and k to four decimal places. A)
B)
C)
D)
E)
Question
Question
A calf that weighs 60 pounds at birth gains weight at the rate 
where w is weight in pounds and t is time in years.What is the maximum weight of the animal if one uses the model 
?
A)1200 lb
B)810 lb
C)1140 lb
D)1260 lb
E)1950 lb
where w is weight in pounds and t is time in years.What is the maximum weight of the animal if one uses the model ? A)1200 lb
B)810 lb
C)1140 lb
D)1260 lb
E)1950 lb
Question
Write and solve the differential equation that models the following verbal statement.Evaluate the solution at the specified value of the independent variable,rounding your answer to four decimal places:
The rate of change of N is proportional to N.When
, 
and when 
, 
.What is the value of N when 
?
A)
B)
C)
D)
E)
The rate of change of N is proportional to N.When
, and when , .What is the value of N when ?
A)
B)
C)
D)
E)
Question
Find the function 
passing through the point 
with the first derivative 
.
A)
B)
C)
D)
E)
passing through the point with the first derivative . A)
B)
C)
D)
E)
Question
Find the orthogonal trajectories of the family 
.
A)
B)
C)
D)
E)
.
A)
B)
C)
D)
E)
Question
Find the orthogonal trajectories of the family 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
The rate of change of N is proportional to N.When 
and when 
.What is the value of N when 
? Round your answer to three decimal places.
A)1,736.667
B)1,706.667
C)1,656.667
D)59.112
E)5,120.000
and when .What is the value of N when ? Round your answer to three decimal places. A)1,736.667
B)1,706.667
C)1,656.667
D)59.112
E)5,120.000
Question
Find the function 
passing through the point 
with the first derivative 
.
A)
B)
C)
D)
E)
passing through the point with the first derivative . A)
B)
C)
D)
E)
Question
Solve the differential equation 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Suppose that the population (in millions)of a Paraguay in 2007 is 6.7 and that expected continuous annual rate of change of the population is 0.024.The exponential growth model for the population by letting 
corresponds to 2000 is 
.Use the model to predict the population of the country in 2015.Round your answer to two decimal places.
A)7.03 million
B)7.93 million
C)8.32 million
D)8.12 million
E)6.86 million
corresponds to 2000 is .Use the model to predict the population of the country in 2015.Round your answer to two decimal places. A)7.03 million
B)7.93 million
C)8.32 million
D)8.12 million
E)6.86 million
Question
Suppose that the population (in millions)of Hungary in 2007 was 10 and that the expected continuous annual rate of change of the population is -0.003.Find the exponential growth model 
for the population by letting 
correspond to 2000.Round your answer to four decimal places.
A)
B)
C)
D)
E)
for the population by letting correspond to 2000.Round your answer to four decimal places. A)
B)
C)
D)
E)
Question
Use integration to find a general solution of the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Which of the following is a solution of the differential equation 
?
A)
B)
C)
D)
E)
? A)
B)
C)
D)
E)
Question
Find the particular solution of the differential equation 
that satisfies the initial condition 
when 
,where 
is the general solution.
A)
B)
C)
D)
E)
that satisfies the initial condition when ,where is the general solution. A)
B)
C)
D)
E)
Question
Question
Use integration to find a general solution of the differential equation 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Question
Use integration to find a general solution of the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Use integration to find a general solution of the differential equation . 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Use integration to find a general solution of the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
A container of hot liquid is placed in a freezer that is kept at a constant temperature of 
.The initial temperature of the liquid is 
.After 5 minutes,the liquid's temperature is 
.How much longer will it take for its temperature to decrease to 
? Round your answer to two decimal places.
A)5.29 minutes
B)7.94 minutes
C)8.82 minutes
D)3.53 minutes
E)9.70 minutes
.The initial temperature of the liquid is .After 5 minutes,the liquid's temperature is .How much longer will it take for its temperature to decrease to ? Round your answer to two decimal places. A)5.29 minutes
B)7.94 minutes
C)8.82 minutes
D)3.53 minutes
E)9.70 minutes
Question
Use integration to find a general solution of the differential equation 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Which of the following is a solution of the differential equation 
?
A)
B)
C)
D)
E)
? A)
B)
C)
D)
E)
Question
The initial investment in a savings account in which interest is compounded continuously is $850.If the time required to double the amount is 
years,what is the amount after 14 years? Round your answer to the nearest cent.
A)$3,400.00
B)$3,099.86
C)$3,173.33
D)$2,699.86
E)$9,197.18
years,what is the amount after 14 years? Round your answer to the nearest cent. A)$3,400.00
B)$3,099.86
C)$3,173.33
D)$2,699.86
E)$9,197.18
Question
The number of bacteria in a culture is increasing according to the law of exponential growth.After 5 hours there are 130 bacteria in the culture and after 10 hours there are 330 bacteria in the culture.Answer the following questions,rounding numerical answers to four decimal places.
(I)Find the initial population.
(II)Write an exponential growth model for the bacteria population.Let t represent time in hours.
(III)Use the model to determine the number of bacteria after 20 hours.
(IV)After how many hours will the bacteria count be 10,000?
A)(I)51.2121 ; (II)
; (III)3,439.1838 ; (IV)30.6439 hr
B)(I)53.3621 ; (II)
; (III)4,813.9010 ; (IV)32.8719 hr
C)(I)51.2121 ; (II)
; (III)2,126.4497 ; (IV)28.3094 hr
D)(I)56.9921 ; (II)
; (III)7,565.3544 ; (IV)34.9831 hr
E)(I)58.5521 ; (II)
; (III)10,217.8179 ; (IV)36.7625 hr
(I)Find the initial population.
(II)Write an exponential growth model for the bacteria population.Let t represent time in hours.
(III)Use the model to determine the number of bacteria after 20 hours.
(IV)After how many hours will the bacteria count be 10,000?
A)(I)51.2121 ; (II)
; (III)3,439.1838 ; (IV)30.6439 hrB)(I)53.3621 ; (II)
; (III)4,813.9010 ; (IV)32.8719 hrC)(I)51.2121 ; (II)
; (III)2,126.4497 ; (IV)28.3094 hrD)(I)56.9921 ; (II)
; (III)7,565.3544 ; (IV)34.9831 hrE)(I)58.5521 ; (II)
; (III)10,217.8179 ; (IV)36.7625 hr Question
The initial investment in a savings account in which interest is compounded continuously is $775.If the time required to double the amount is 
years,what is the annual rate? Round your answer to two decimal places.
A)9.24 %
B)9.90 %
C)10.82 %
D)8.04 %
E)10.64 %
years,what is the annual rate? Round your answer to two decimal places. A)9.24 %
B)9.90 %
C)10.82 %
D)8.04 %
E)10.64 %
Question
Which of the following is a solution of the differential equation 
?
A)
B)
C)
D)
E)
? A)
B)
C)
D)
E)
Question
Question
Use integration to find a general solution of the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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Deck 6: Differential Equations
1
A population of rabbits in a certain habitat grows according to the differential equation where t is measured in months and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is, .
Estimates of y(t)can be produced using Euler's Method with step size .To the nearest rabbit,the estimate for y(2)is
A)176
B)358
C)353
D)349
where t is measured in months and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is, . Estimates of y(t)can be produced using Euler's Method with step size
.To the nearest rabbit,the estimate for y(2)is
A)176
B)358
C)353
D)349
353
2
Water flows continuously from a large tank at a rate proportional to the amount of water remaining in the tank;that is, .There was initially 10,000 cubic feet of water in the tank,and at time hours,6,000 cubic feet of water remained.
To the nearest cubic foot,how much water remained in the tank at time hours?
A)4,937
B)5,170
C)5,394
D)4,891
.There was initially 10,000 cubic feet of water in the tank,and at time hours,6,000 cubic feet of water remained. To the nearest cubic foot,how much water remained in the tank at time
hours?
A)4,937
B)5,170
C)5,394
D)4,891
4,891
3
Consider the differential equation ,for only,with initial value .
Which of the following is the slope field for the general solution to the given differential equation?
A)
B)
C)
D)
,for only,with initial value .
Which of the following is the slope field for the general solution to the given differential equation?
A)
B)
C)
D)
4
A population of rabbits in a certain habitat grows according to the differential equation where t is measured in months and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is, .
What is the fastest growth rate,in rabbits per month,that this population exhibits?
A)150
B)600
C)300
D)250
where t is measured in months and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is, . What is the fastest growth rate,in rabbits per month,that this population exhibits?
A)150
B)600
C)300
D)250
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5
Consider the differential equation ,for which the solution is g(x). Let y(x)be the particular solution that contains (0,1).Using Euler's Method with step size ,what is the estimate for ?
A)4
B)3.5
C)3
D)4.5
,for which the solution is g(x). Let y(x)be the particular solution that contains (0,1).Using Euler's Method with step size ,what is the estimate for ?
A)4
B)3.5
C)3
D)4.5
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6
Each of the following graphs is from a logistic function .Which one has the smallest value of b?
A)
B)
C)
D)
E)
.Which one has the smallest value of b? A)
B)
C)
D)
E)
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7
Water is being pumped continuously into a tank at a rate that is inversely proportional to the amount of water in the tank;that is, ,where y is the number of gallons of water in the tank after t minutes .Initially there were 5 gallons of water in the tank,and after 3 inutes there were 8 gallons.How many gallons of water were in the tank at time 20 minutes?
A)
B)
C)
D)
,where y is the number of gallons of water in the tank after t minutes .Initially there were 5 gallons of water in the tank,and after 3 inutes there were 8 gallons.How many gallons of water were in the tank at time 20 minutes? A)
B)
C)
D)
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8
Consider the differential equation ,for which the solution is g(x). Which of the following statements is true about the particular solution that contains (1,-4)?
A)g(x)is increasing and concave up.
B)g(x)is increasing and concave down.
C)g(x)is decreasing and concave up.
D)g(x)is decreasing and concave down.
,for which the solution is g(x). Which of the following statements is true about the particular solution that contains (1,-4)?
A)g(x)is increasing and concave up.
B)g(x)is increasing and concave down.
C)g(x)is decreasing and concave up.
D)g(x)is decreasing and concave down.
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9
The general solution to the differential equation is
A)
B)
C)
D)
is A)
B)
C)
D)
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10
If then the particular solution is
A)
B)
C)
D)
then the particular solution is A)
B)
C)
D)
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11
For what value(s)of x is ?
? Unlock Deck
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12
Solve the differential equation in to find f(x)in terms of x only.
to find f(x)in terms of x only. Unlock Deck
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13
Consider the differential equation ,for only,with initial value .
The particular solution y(x)is
A)
B)
C)
D)
,for only,with initial value .
The particular solution y(x)is
A)
B)
C)
D)
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14
A population of rabbits in a certain habitat grows according to the differential equation where t is measured in months and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is, .
Which of the following slope fields represents an approximate general solution to the given differential equation?
A)
B)
C)
D)
where t is measured in months and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is, . Which of the following slope fields represents an approximate general solution to the given differential equation?
A)
B)
C)
D)
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15
Identify the graph of the logistic function .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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16
Consider the differential equation ,for which the solution is .Let . The particular solution is
A)
B)
C)
D)
,for which the solution is .Let . The particular solution is
A)
B)
C)
D)
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17
A differentiable function is defined such that,for all values of in its domain, .
What is the domain of ?
A differentiable function
is defined such that,for all values of in its domain, .What is the domain of
? Unlock Deck
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18
Water flows continuously from a large tank at a rate proportional to the amount of water remaining in the tank;that is, .There was initially 9,000 cubic feet of water in the tank,and at time hours,8,000 cubic feet of water remained.
What is the value of k in the equation ? Round your answer to 3 decimal places.
A)0.014
B)0.060
C)-0.131
D)-0.059
.There was initially 9,000 cubic feet of water in the tank,and at time hours,8,000 cubic feet of water remained. What is the value of k in the equation
? Round your answer to 3 decimal places.
A)0.014
B)0.060
C)-0.131
D)-0.059
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19
Show that .
. Unlock Deck
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20
Consider the differential equation ,for only,with initial value .
Using Euler's Method with step size ,what is the estimate for ?
A)
B)2.5
C)3
D)4.85
,for only,with initial value .
Using Euler's Method with step size
,what is the estimate for ?
A)
B)2.5
C)3
D)4.85
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21
The logistic function models the growth of a population.Identify the initial population.
A)10
B)12
C)5
D)60
E)6
models the growth of a population.Identify the initial population. A)10
B)12
C)5
D)60
E)6
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22
Match the logistic differential equation and initial condition with the graph of its solution shown below.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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23
Find the particular solution of the differential equation that satisfies the initial condition .
A)
B)
C)
D)
E)
that satisfies the initial condition . A)
B)
C)
D)
E)
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24
Each of the following graphs is from a logistic function .Which one has the largest value of b?
A)
B)
C)
D)
E)
.Which one has the largest value of b? A)
B)
C)
D)
E)
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25
The logistic function models the growth of a population.Determine when the population reaches 40% of the maximum carrying capacity.Round your answer to three decimal places.
A)4.317
B)3.000
C)0.474
D)0.677
E)0.301
models the growth of a population.Determine when the population reaches 40% of the maximum carrying capacity.Round your answer to three decimal places. A)4.317
B)3.000
C)0.474
D)0.677
E)0.301
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26
A conservation organization releases 30 panthers into a preserve.After 3 years,there are 50 panthers in the preserve.The preserve has a carrying capacity of 150.Write a logistic function that models the population of panthers in the preserve.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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27
Determine whether the function is homogeneous and determine its degree if it is.
A)homogeneous,the degree is 3
B)homogeneous,the degree is 4
C)not homogeneous00
D)homogeneous,the degree is 2
E)homogeneous,the degree is 1
is homogeneous and determine its degree if it is. A)homogeneous,the degree is 3
B)homogeneous,the degree is 4
C)not homogeneous00
D)homogeneous,the degree is 2
E)homogeneous,the degree is 1
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28
Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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29
Match the logistic equation and initial condition with the graph of the solution.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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30
A conservation organization releases 40 coyotes into a preserve.After 4 years,there are 70 coyotes in the preserve.The preserve has a carrying capacity of 175.Determine the time it takes for the population to reach 140.
A)15.599 years
B)11.454 years
C)12.838 years
D)10.271 years
E)14.463 years
A)15.599 years
B)11.454 years
C)12.838 years
D)10.271 years
E)14.463 years
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31
Find the logistic equation that satisfies the following differential equation and initial condition.
A)
B)
C)
D)
E)none of the above
A)
B)
C)
D)
E)none of the above
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32
The logistic function models the growth of a population.Determine when the population reaches one-half of the maximum carrying capacity.Round your answer to three decimal places.
A)0.549
B)3.333
C)1.151
D)5.000
E)1.000
models the growth of a population.Determine when the population reaches one-half of the maximum carrying capacity.Round your answer to three decimal places. A)0.549
B)3.333
C)1.151
D)5.000
E)1.000
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33
Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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34
A calf that weighs 75 pounds at birth gains weight at the rate where w is weight in pounds and t is time in years.Use a computer algebra system to solve the differential equation for .
A)
B)
C)
D)
E)
where w is weight in pounds and t is time in years.Use a computer algebra system to solve the differential equation for . A)
B)
C)
D)
E)
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35
The logistic function models the growth of a population.Identify the value of k.
A)3.5
B)4
C)2
D)10
E)3
models the growth of a population.Identify the value of k. A)3.5
B)4
C)2
D)10
E)3
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36
Find the general solution of the differential equation .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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37
Find an equation of the graph that passes through the point (2,5)and has the slope .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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38
A conservation organization releases 30 panthers into a preserve.After 3 years,there are 50 panthers in the preserve.The preserve has a carrying capacity of 150.Determine the population after 6 years.Discard any fractional part of your answer.
A)74
B)66
C)87
D)79
E)130
A)74
B)66
C)87
D)79
E)130
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39
Solve the homogeneous differential equation .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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40
The logistic function models the growth of a population.Identify the maximum carrying capacity.
A)15
B)5.5
C)5
D)4
E)6
models the growth of a population.Identify the maximum carrying capacity. A)15
B)5.5
C)5
D)4
E)6
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41
Write and solve the differential equation that models the following verbal statement:
The rate of change of M with respect to s is proportional to .
A)
B)
C)
D)
E)
The rate of change of M with respect to s is proportional to
.
A)
B)
C)
D)
E)
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42
The isotope has a half-life of 24,100 years.After 1,000 years,a sample of the isotope is reduced to 1.4 grams.What was the initial size of the sample (in grams)? How much will remain after 10,000 years (i.e. ,after another 9,000 years)? Round your answers to four decimal places.
A)1.0086 g,0.7565 g
B)2.3054 g,1.7291 g
C)1.4409 g,1.0807 g
D)2.0172 g,1.5130 g
E)1.8731 g,1.4049 g
has a half-life of 24,100 years.After 1,000 years,a sample of the isotope is reduced to 1.4 grams.What was the initial size of the sample (in grams)? How much will remain after 10,000 years (i.e. ,after another 9,000 years)? Round your answers to four decimal places. A)1.0086 g,0.7565 g
B)2.3054 g,1.7291 g
C)1.4409 g,1.0807 g
D)2.0172 g,1.5130 g
E)1.8731 g,1.4049 g
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43
Solve the differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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44
The isotope has a half-life of 5,715 years.After 20,000 years,a sample of the isotope is reduced 0.7 grams.What was the initial size of the sample (in grams)? How large was the sample after the first 2,000 years? Round your answers to four decimal places.
A)7.9172 g,6.2119 g
B)10.2924 g,8.0755 g
C)4.7503 g,3.7271 g
D)6.3338 g,4.9695 g
E)3.9586 g,3.1059 g
has a half-life of 5,715 years.After 20,000 years,a sample of the isotope is reduced 0.7 grams.What was the initial size of the sample (in grams)? How large was the sample after the first 2,000 years? Round your answers to four decimal places. A)7.9172 g,6.2119 g
B)10.2924 g,8.0755 g
C)4.7503 g,3.7271 g
D)6.3338 g,4.9695 g
E)3.9586 g,3.1059 g
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45
Solve the differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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46
Solve the differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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47
Solve the differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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48
A calf that weighs 60 pounds at birth gains weight at the rate where w is weight in pounds and t is time in years.If the animal is sold when its weight reaches 650 pounds,find the time of sale using the model .Round your answer to two decimal places.
A)7.31 years
B)2.65 years
C)1.07 years
D)0.72 year
E)0.29 year
where w is weight in pounds and t is time in years.If the animal is sold when its weight reaches 650 pounds,find the time of sale using the model .Round your answer to two decimal places. A)7.31 years
B)2.65 years
C)1.07 years
D)0.72 year
E)0.29 year
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49
The half-life of the carbon isotope C-14 is approximately 5,715 years.If the amount left after 3,000 years is 1.8 g,what is the amount after 6,000 years? Round your answer to three decimal places.
A)1.251 g
B)1.043 g
C)1.800 g
D)0.604 g
E)2.502 g
A)1.251 g
B)1.043 g
C)1.800 g
D)0.604 g
E)2.502 g
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50
The isotope has a half-life of 24,100 years.Given an initial amount of 6 grams of the isotope,how many grams will remain after 500 years? After 5,000 years? Round your answers to four decimal places.
A)4.1400 g,3.6374 g
B)3.5486 g,3.1178 g
C)5.9143 g,5.1963 g
D)2.3657 g,2.0785 g
E)7.0972 g,6.2356 g
has a half-life of 24,100 years.Given an initial amount of 6 grams of the isotope,how many grams will remain after 500 years? After 5,000 years? Round your answers to four decimal places. A)4.1400 g,3.6374 g
B)3.5486 g,3.1178 g
C)5.9143 g,5.1963 g
D)2.3657 g,2.0785 g
E)7.0972 g,6.2356 g
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51
Find the exponential function that passes through the two given points.Round your values of C and k to four decimal places.
A)
B)
C)
D)
E)
that passes through the two given points.Round your values of C and k to four decimal places. A)
B)
C)
D)
E)
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52
The half-life of the carbon isotope C-14 is approximately 5,715 years.If the initial quantity of the isotope is 38 g,what is the amount left after 1,000 years? Round your answer to two decimal places.
A)33.66 g
B)35.76 g
C)34.16 g
D)3.32 g
E)16.83 g
A)33.66 g
B)35.76 g
C)34.16 g
D)3.32 g
E)16.83 g
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53
A calf that weighs 60 pounds at birth gains weight at the rate where w is weight in pounds and t is time in years.What is the maximum weight of the animal if one uses the model ?
A)1200 lb
B)810 lb
C)1140 lb
D)1260 lb
E)1950 lb
where w is weight in pounds and t is time in years.What is the maximum weight of the animal if one uses the model ? A)1200 lb
B)810 lb
C)1140 lb
D)1260 lb
E)1950 lb
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54
Write and solve the differential equation that models the following verbal statement.Evaluate the solution at the specified value of the independent variable,rounding your answer to four decimal places:
The rate of change of N is proportional to N.When , and when , .What is the value of N when ?
A)
B)
C)
D)
E)
The rate of change of N is proportional to N.When
, and when , .What is the value of N when ?
A)
B)
C)
D)
E)
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55
Find the function passing through the point with the first derivative .
A)
B)
C)
D)
E)
passing through the point with the first derivative . A)
B)
C)
D)
E)
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56
Find the orthogonal trajectories of the family .
A)
B)
C)
D)
E)
.
A)
B)
C)
D)
E)
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57
Find the orthogonal trajectories of the family .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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58
The rate of change of N is proportional to N.When and when .What is the value of N when ? Round your answer to three decimal places.
A)1,736.667
B)1,706.667
C)1,656.667
D)59.112
E)5,120.000
and when .What is the value of N when ? Round your answer to three decimal places. A)1,736.667
B)1,706.667
C)1,656.667
D)59.112
E)5,120.000
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59
Find the function passing through the point with the first derivative .
A)
B)
C)
D)
E)
passing through the point with the first derivative . A)
B)
C)
D)
E)
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60
Solve the differential equation .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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61
Suppose that the population (in millions)of a Paraguay in 2007 is 6.7 and that expected continuous annual rate of change of the population is 0.024.The exponential growth model for the population by letting corresponds to 2000 is .Use the model to predict the population of the country in 2015.Round your answer to two decimal places.
A)7.03 million
B)7.93 million
C)8.32 million
D)8.12 million
E)6.86 million
corresponds to 2000 is .Use the model to predict the population of the country in 2015.Round your answer to two decimal places. A)7.03 million
B)7.93 million
C)8.32 million
D)8.12 million
E)6.86 million
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62
Suppose that the population (in millions)of Hungary in 2007 was 10 and that the expected continuous annual rate of change of the population is -0.003.Find the exponential growth model for the population by letting correspond to 2000.Round your answer to four decimal places.
A)
B)
C)
D)
E)
for the population by letting correspond to 2000.Round your answer to four decimal places. A)
B)
C)
D)
E)
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63
Use integration to find a general solution of the differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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64
Which of the following is a solution of the differential equation ?
A)
B)
C)
D)
E)
? A)
B)
C)
D)
E)
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65
Find the particular solution of the differential equation that satisfies the initial condition when ,where is the general solution.
A)
B)
C)
D)
E)
that satisfies the initial condition when ,where is the general solution. A)
B)
C)
D)
E)
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66
The half-life of the radium isotope Ra-226 is approximately 1,599 years.What percent of a given amount remains after 100 years? Round your answer to two decimal places.
A)95.76 %
B)5.96 %
C)97.76 %
D)3.13 %
E)0.96 %
A)95.76 %
B)5.96 %
C)97.76 %
D)3.13 %
E)0.96 %
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67
Use integration to find a general solution of the differential equation
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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68
Find the principal that must be invested at the rate 8%,compounded monthly,so that $3,000,000 will be available for retirement in 50 years.Round your answer to the nearest cent.
A)$750,000.00
B)$55,681.16
C)$2,151,973.12
D)$1,000,000.00
E)$63,963.69
A)$750,000.00
B)$55,681.16
C)$2,151,973.12
D)$1,000,000.00
E)$63,963.69
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69
Use integration to find a general solution of the differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 88 flashcards in this deck.
Unlock Deck
k this deck
70
Use integration to find a general solution of the differential equation .
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 88 flashcards in this deck.
Unlock Deck
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71
Use integration to find a general solution of the differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 88 flashcards in this deck.
Unlock Deck
k this deck
72
A container of hot liquid is placed in a freezer that is kept at a constant temperature of .The initial temperature of the liquid is .After 5 minutes,the liquid's temperature is .How much longer will it take for its temperature to decrease to ? Round your answer to two decimal places.
A)5.29 minutes
B)7.94 minutes
C)8.82 minutes
D)3.53 minutes
E)9.70 minutes
.The initial temperature of the liquid is .After 5 minutes,the liquid's temperature is .How much longer will it take for its temperature to decrease to ? Round your answer to two decimal places. A)5.29 minutes
B)7.94 minutes
C)8.82 minutes
D)3.53 minutes
E)9.70 minutes
Unlock Deck
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73
Use integration to find a general solution of the differential equation .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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74
Which of the following is a solution of the differential equation ?
A)
B)
C)
D)
E)
? A)
B)
C)
D)
E)
Unlock Deck
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Unlock Deck
k this deck
75
The initial investment in a savings account in which interest is compounded continuously is $850.If the time required to double the amount is years,what is the amount after 14 years? Round your answer to the nearest cent.
A)$3,400.00
B)$3,099.86
C)$3,173.33
D)$2,699.86
E)$9,197.18
years,what is the amount after 14 years? Round your answer to the nearest cent. A)$3,400.00
B)$3,099.86
C)$3,173.33
D)$2,699.86
E)$9,197.18
Unlock Deck
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76
The number of bacteria in a culture is increasing according to the law of exponential growth.After 5 hours there are 130 bacteria in the culture and after 10 hours there are 330 bacteria in the culture.Answer the following questions,rounding numerical answers to four decimal places.
(I)Find the initial population.
(II)Write an exponential growth model for the bacteria population.Let t represent time in hours.
(III)Use the model to determine the number of bacteria after 20 hours.
(IV)After how many hours will the bacteria count be 10,000?
A)(I)51.2121 ; (II) ; (III)3,439.1838 ; (IV)30.6439 hr
B)(I)53.3621 ; (II) ; (III)4,813.9010 ; (IV)32.8719 hr
C)(I)51.2121 ; (II) ; (III)2,126.4497 ; (IV)28.3094 hr
D)(I)56.9921 ; (II) ; (III)7,565.3544 ; (IV)34.9831 hr
E)(I)58.5521 ; (II) ; (III)10,217.8179 ; (IV)36.7625 hr
(I)Find the initial population.
(II)Write an exponential growth model for the bacteria population.Let t represent time in hours.
(III)Use the model to determine the number of bacteria after 20 hours.
(IV)After how many hours will the bacteria count be 10,000?
A)(I)51.2121 ; (II)
; (III)3,439.1838 ; (IV)30.6439 hrB)(I)53.3621 ; (II)
; (III)4,813.9010 ; (IV)32.8719 hrC)(I)51.2121 ; (II)
; (III)2,126.4497 ; (IV)28.3094 hrD)(I)56.9921 ; (II)
; (III)7,565.3544 ; (IV)34.9831 hrE)(I)58.5521 ; (II)
; (III)10,217.8179 ; (IV)36.7625 hr Unlock Deck
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77
The initial investment in a savings account in which interest is compounded continuously is $775.If the time required to double the amount is years,what is the annual rate? Round your answer to two decimal places.
A)9.24 %
B)9.90 %
C)10.82 %
D)8.04 %
E)10.64 %
years,what is the annual rate? Round your answer to two decimal places. A)9.24 %
B)9.90 %
C)10.82 %
D)8.04 %
E)10.64 %
Unlock Deck
Unlock for access to all 88 flashcards in this deck.
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78
Which of the following is a solution of the differential equation ?
A)
B)
C)
D)
E)
? A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 88 flashcards in this deck.
Unlock Deck
k this deck
79
Find the time (in years)necessary for 1,000 to double if it is invested at a rate 6% compounded continuously.Round your answer to two decimal places.
A)1.16 years
B)11.55 years
C)1.39 years
D)11.90 years
E)11.58 years
A)1.16 years
B)11.55 years
C)1.39 years
D)11.90 years
E)11.58 years
Unlock Deck
Unlock for access to all 88 flashcards in this deck.
Unlock Deck
k this deck
80
Use integration to find a general solution of the differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 88 flashcards in this deck.
Unlock Deck
k this deck
Unlock Deck
Unlock for access to all 88 flashcards in this deck.
