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Deck 6: Differential Equations
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Question
At time 
minutes, the temperature of an object is 
F. The temperature of the object is changing at the rate given by the differential equation 
. Use Euler's Method to approximate the particular solutions of this differential equation at 
. Use a step size of 
. Round your answer to one decimal place.
A) 137.1
B) 139.5
C) 147.4
D) 144.6
E) 132.7
minutes, the temperature of an object is F. The temperature of the object is changing at the rate given by the differential equation . Use Euler's Method to approximate the particular solutions of this differential equation at . Use a step size of . Round your answer to one decimal place. A) 137.1
B) 139.5
C) 147.4
D) 144.6
E) 132.7
Question
Use integration to find a general solution of the differential equation 
.
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 homogeneous
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 homogeneous
D) homogeneous, the degree is 2
E) homogeneous, the degree is 1
Question
Question
The isotope 
has a half-life of 
years. Given an initial amount of 16 grams of the isotope, how many grams will remain after 1,000 years? After 10,000 years? Round your answers to four decimal places.
A) 10.8825 g, 8.4006 g
B) 9.3278 g, 7.2005 g
C) 15.5464 g, 12.0008 g
D) 6.2185 g, 4.8003 g
E) 18.6556 g, 14.4010 g
has a half-life of years. Given an initial amount of 16 grams of the isotope, how many grams will remain after 1,000 years? After 10,000 years? Round your answers to four decimal places. A) 10.8825 g, 8.4006 g
B) 9.3278 g, 7.2005 g
C) 15.5464 g, 12.0008 g
D) 6.2185 g, 4.8003 g
E) 18.6556 g, 14.4010 g
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
A calf that weighs 50 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) 750 lb
C) 1150 lb
D) 1250 lb
E) 1900 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) 750 lb
C) 1150 lb
D) 1250 lb
E) 1900 lb
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) 1,316.250
B) 1,286.250
C) 1,236.250
D) 140.599
E) 20,580.000
, and when , . What is the value of N when ? Round your answer to three decimal places. A) 1,316.250
B) 1,286.250
C) 1,236.250
D) 140.599
E) 20,580.000
Question
Solve the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
A 500-gallon tank is half full of distilled water. At time 
, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 11 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 9 gallons per minute. At what time will the tank be full?
A) 250 minutes
B) 126 minutes
C) 125 minutes
D) 501 minutes
E) 1000 minutes
, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 11 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 9 gallons per minute. At what time will the tank be full? A) 250 minutes
B) 126 minutes
C) 125 minutes
D) 501 minutes
E) 1000 minutes
Question
Solve the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
A 600-gallon tank is full of a solution containing 55 pounds of concentrate. Starting at time 
distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the amount of concentrate Q in the solution as a function of t.
A)
B)
C)
D)
E)
distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the amount of concentrate Q in the solution as a function of t.A)
B)
C)
D)
E)
Question
Find an equation of the graph that passes through the point 
and has the slope 
.
A)
B)
C)
D)
E)
and has the slope . A)
B)
C)
D)
E)
Question
Solve the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Question
Sketch the slope field for the differential equation 
and use the slope field to sketch the solution satisfying the condition 
.
A)
B)
C)
D)
E)
and use the slope field to sketch the solution satisfying the condition . A)
B)
C)
D)
E)
Question
Solve the first order linear differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
The logistic function 
models the growth of a population. Determine when the population reaches 
% 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 % 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
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
Question
A 200-gallon tank is full of a solution containing 25 pounds of concentrate. Starting at time 
distilled water is added to the tank at a rate of 20 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the time at which the amount of concentrate in the tank reaches 15 pounds. Round your answer to one decimal place.
A) 2.2 min
B) 1.8 min
C) 10.2 min
D) 5.1 min
E) 10.2 min
distilled water is added to the tank at a rate of 20 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the time at which the amount of concentrate in the tank reaches 15 pounds. Round your answer to one decimal place.A) 2.2 min
B) 1.8 min
C) 10.2 min
D) 5.1 min
E) 10.2 min
Question
Sketch the slope field for the differential equation 
and use the slope field to sketch the solution that passes through the point 
.
A)
B)
C)
D)
E)
and use the slope field to sketch the solution that passes through the point . A)
B)
C)
D)
E)
Question
Find the particular solution of the differential equation 
that satisfies the boundary condition 
.
A)
B)
C)
D)
E)
that satisfies the boundary condition . A)
B)
C)
D)
E)
Question
Find the orthogonal trajectories of the family 
.
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
Find the particular solution of the differential equation 
that satisfies the boundary condition 
.
A)
B)
C)
D)
E)
that satisfies the boundary condition . A)
B)
C)
D)
E)
Question
Select from the choices below the slope field for the differential equation. 
A)
B)
C)
D)
E) none of the above
A)
B)
C)
D)
E) none of the above
Question
Use integration to find a general solution of the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Identify the graph of the logistic function 
.
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
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
Suppose that the population (in millions) of a Uganda in 2007 is 30.3 and that expected continuous annual rate of change of the population is 0.036. The exponential growth model for the population by letting 
corresponds to 2000 is 
. Use the model to predict the population of the country in 2014. Round your answer to two decimal places.
A) 32.56 millions
B) 37.61 millions
C) 40.41 millions
D) 38.98 millions
E) 31.41 millions
corresponds to 2000 is . Use the model to predict the population of the country in 2014. Round your answer to two decimal places. A) 32.56 millions
B) 37.61 millions
C) 40.41 millions
D) 38.98 millions
E) 31.41 millions
Question
The isotope 
has a half-life of 1,599 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) 4076.8643 g, 1713.1817 g
B) 5299.9236 g, 2227.1362 g
C) 2446.1186 g, 1027.9090 g
D) 3261.4914 g, 1370.5454 g
E) 2038.4321 g, 856.5909 g
has a half-life of 1,599 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) 4076.8643 g, 1713.1817 g
B) 5299.9236 g, 2227.1362 g
C) 2446.1186 g, 1027.9090 g
D) 3261.4914 g, 1370.5454 g
E) 2038.4321 g, 856.5909 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
The logistic function 
models the growth of a population. Identify the initial population.
A) 6
B) 8
C) 3
D) 24
E) 2
models the growth of a population. Identify the initial population.A) 6
B) 8
C) 3
D) 24
E) 2
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 logistic function 
models the growth of a population. Identify the value of k.
A) 1.7
B) 2.2
C) 0.2
D) 20
E) 0.5
models the growth of a population. Identify the value of k.A) 1.7
B) 2.2
C) 0.2
D) 20
E) 0.5
Question
Solve 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
Solve the Bernoulli differential equation 
.
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
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
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
Question
The initial investment in a savings account in which interest is compounded continuously is $803. If the time required to double the amount is 
years, what is the annual rate? Round your answer to two decimal places.
A) 7.30 %
B) 7.70 %
C) 13.71 %
D) 6.10 %
E) 8.70 %
years, what is the annual rate? Round your answer to two decimal places. A) 7.30 %
B) 7.70 %
C) 13.71 %
D) 6.10 %
E) 8.70 %
Question
Use integration to find a general solution of the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Assume an object weighing 7 pounds is dropped from a height of 9,000 feet, where the air resistance is proportional to the velocity. Round numerical answers in your answer to two places.
(i) Write the velocity as a function of time if the object's velocity after 4 seconds is 2.33 feet per second.
(ii) What is the limiting value of the velocity function?
A) (i)
; (ii) 0
B) (i)
; (ii) 0
C) (i)
; (ii) 2.33
D) (i)
; (ii) 2.33
E) (i)
; (ii) limit does not exist
(i) Write the velocity as a function of time if the object's velocity after 4 seconds is 2.33 feet per second.
(ii) What is the limiting value of the velocity function?
A) (i)
; (ii) 0B) (i)
; (ii) 0C) (i)
; (ii) 2.33D) (i)
; (ii) 2.33E) (i)
; (ii) limit does not exist Question
Find an equation of the graph that passes through the point 
and has the slope 
.
A)
B)
C)
D)
E)
and has the slope . 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
Use Euler's Method to make a table of values for the approximate solution of the following differential equation with specified initial value. Use 5 steps of size 0.05. 
, 
A) 
0)000
1)000
2)000
3)000
4)000
5)000
B) 
0)000
1)000
2)000
3)000
4)000
5)000
C) 
0)000
1)000
2)000
3)000
4)000
5)000
D) 
0)000
1)000
2)000
3)000
4)000
5)000
E) 
0)000
1)000
2)000
3)000
4)000
5)000
, A)
0)000
1)000
2)000
3)000
4)000
5)000
B)
0)000
1)000
2)000
3)000
4)000
5)000
C)
0)000
1)000
2)000
3)000
4)000
5)000
D)
0)000
1)000
2)000
3)000
4)000
5)000
E)
0)000
1)000
2)000
3)000
4)000
5)000
Question
Find the logistic equation that satisfies the following differential equation and initial condition. 
, 
A)
B)
C)
D)
E) none of these
, A)
B)
C)
D)
E) none of these
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. If the animal is sold when its weight reaches 900 pounds, find the time of sale using the model 
. Round your answer to two decimal places.
A) 7.02 years
B) 2.76 years
C) 1.12 years
D) 1.08 years
E) 0.44 year
where w is weight in pounds and t is time in years. If the animal is sold when its weight reaches 900 pounds, find the time of sale using the model . Round your answer to two decimal places. A) 7.02 years
B) 2.76 years
C) 1.12 years
D) 1.08 years
E) 0.44 year
Question
Solve the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Question
Use the differential equation 
and its slope field to find the slope at the point 
. 
A) -8
B) -1
C) -4
D) -16
E) 8
and its slope field to find the slope at the point . A) -8
B) -1
C) -4
D) -16
E) 8
Question
Select from the choices below the slope field for the differential equation. 
A)
B)
C)
D) none of the above
A)
B)
C)
D) none of the above
Question
Find the particular solution of the differential equation 
that satisfies the boundary condition 
.
A)
B)
C)
D)
E)
that satisfies the boundary condition . A)
B)
C)
D)
E)
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
A 300-gallon tank is half full of distilled water. At time 
, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 6 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 4 gallons per minute. At the time the tank is full, how many pounds of concentrate will it contain? Round your answer to two decimal places.
A)
lbs
B)
lbs
C)
lbs
D)
lbs
E)
lbs
, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 6 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 4 gallons per minute. At the time the tank is full, how many pounds of concentrate will it contain? Round your answer to two decimal places.A)
lbsB)
lbsC)
lbsD)
lbsE)
lbs Question
Solve the first order linear differential equation. 
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 number of bacteria in a culture is increasing according to the law of exponential growth. After 2 hours there are 135 bacteria in the culture and after 4 hours there are 390 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 8 hours.
(iv) After how many hours will the bacteria count be 25,000?
A) (i) 46.7341; (ii)
; (iii) 4,566.8441; (iv) 14.1787 hr
B) (i) 48.8841; (ii)
; (iii) 5,941.5613; (iv) 16.4067 hr
C) (i) 46.7341; (ii)
; (iii) 3,254.11; (iv) 11.8442 hr
D) (i) 52.5141; (ii)
; (iii) 8,693.0147; (iv) 18.5179hr
E) (i) 54.0741; (ii)
; (iii) 11,345.4782; (iv) 20.2973 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 8 hours.
(iv) After how many hours will the bacteria count be 25,000?
A) (i) 46.7341; (ii)
; (iii) 4,566.8441; (iv) 14.1787 hrB) (i) 48.8841; (ii)
; (iii) 5,941.5613; (iv) 16.4067 hrC) (i) 46.7341; (ii)
; (iii) 3,254.11; (iv) 11.8442 hrD) (i) 52.5141; (ii)
; (iii) 8,693.0147; (iv) 18.5179hrE) (i) 54.0741; (ii)
; (iii) 11,345.4782; (iv) 20.2973 hr Question
Use integration to find a general solution of the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the general solution of the differential equation 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
A 300-gallon tank is full of a solution containing 55 pounds of concentrate. Starting at time 
distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the quantity of the concentrate in the solution as 
.
A) 30
B) 56
C) 55
D) 0
E) 1
distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the quantity of the concentrate in the solution as . A) 30
B) 56
C) 55
D) 0
E) 1
Question
The initial investment in a savings account in which interest is compounded continuously is $768. If the time required to double the amount is 
years, what is the amount after 13 years? Round your answer to the nearest cent.
A) $2,090.17
B) $1,982.88
C) $2,101.89
D) $1,582.88
E) $10,525.90
years, what is the amount after 13 years? Round your answer to the nearest cent. A) $2,090.17
B) $1,982.88
C) $2,101.89
D) $1,582.88
E) $10,525.90
Question
The isotope 
has a half-life of 5,715 years. After 2,000 years, a sample of the isotope is reduced to 2.1 grams. What was the initial size of the sample (in grams)? How much will remain after 20,000 years (i.e., after another 18,000 years)? Round your answers to four decimal places.
A) 1.8735 g, 0.1656 g
B) 4.2824 g, 0.3786 g
C) 2.6765 g, 0.2366 g
D) 3.7471 g, 0.3313 g
E) 3.4794 g, 0.3076 g
has a half-life of 5,715 years. After 2,000 years, a sample of the isotope is reduced to 2.1 grams. What was the initial size of the sample (in grams)? How much will remain after 20,000 years (i.e., after another 18,000 years)? Round your answers to four decimal places. A) 1.8735 g, 0.1656 g
B) 4.2824 g, 0.3786 g
C) 2.6765 g, 0.2366 g
D) 3.7471 g, 0.3313 g
E) 3.4794 g, 0.3076 g
Question
The logistic function 
models the growth of a population. Identify the maximum carrying capacity.
A) 10
B) 3.5
C) 3
D) 2
E) 4
models the growth of a population. Identify the maximum carrying capacity.A) 10
B) 3.5
C) 3
D) 2
E) 4
Question
A calf that weighs 70 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
A conservation organization releases 
into a preserve. After 
years, there are 
in the preserve. The preserve has a carrying capacity of 
. Write a logistic function that models the population of 
in the preserve.
A)
B)
C)
D)
E)
into a preserve. After years, there are in the preserve. The preserve has a carrying capacity of . Write a logistic function that models the population of in the preserve.A)
B)
C)
D)
E)
Question
Select from the choices below the slope field for the differential equation. 
A)
B)
C)
D)
E) none of the above
A)
B)
C)
D)
E) none of the above
Question
Use 
as a integrating factor to find the general solution of the differential equation 
.
A)
B)
C)
D)
E)
as a integrating factor to find the general solution of the differential equation . 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
Find the general solution of the differential equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Question
A phase trajectory is shown for populations of rabbits and foxes. Describe how each population changes as time goes by. 
Select the correct statement.
A) At
the population of foxes reaches a minimum of about 30.
B) At
the number of rabbits rebounds to 500.
C) At
the number of foxes reaches a maximum of about 2400.
Select the correct statement.
A) At
the population of foxes reaches a minimum of about 30.B) At
the number of rabbits rebounds to 500.C) At
the number of foxes reaches a maximum of about 2400. Question
Write and solve the differential equation that models the following verbal statement:
The rate of change of
with respect to 
is proportional to 
.
A)
, 
B)
, 
C)
, 
D)
, 
E)
, 
The rate of change of
with respect to is proportional to .
A)
, B)
, C)
, D)
, E)
, Question
Find the orthogonal trajectories of the family 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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
Suppose an eight-pound object is dropped from a height of 5,000 feet, where the air resistance is proportional to the velocity. Write the velocity as a function of time if its velocity after 4 seconds is approximately -50 feet per second. Use a graphing utility or a computer algebra system. Round numerical answers in your answer to four places.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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Deck 6: Differential Equations
1
At time minutes, the temperature of an object is F. The temperature of the object is changing at the rate given by the differential equation . Use Euler's Method to approximate the particular solutions of this differential equation at . Use a step size of . Round your answer to one decimal place.
A) 137.1
B) 139.5
C) 147.4
D) 144.6
E) 132.7
minutes, the temperature of an object is F. The temperature of the object is changing at the rate given by the differential equation . Use Euler's Method to approximate the particular solutions of this differential equation at . Use a step size of . Round your answer to one decimal place. A) 137.1
B) 139.5
C) 147.4
D) 144.6
E) 132.7
E
2
Use integration to find a general solution of the differential equation .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
A
3
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 homogeneous
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 homogeneous
D) homogeneous, the degree is 2
E) homogeneous, the degree is 1
C
4
The half-life of the carbon isotope C-14 is approximately 5,715 years. If the initial quantity of the isotope is 34 g, what is the amount left after 10,000 years? Round your answer to two decimal places.
A) 10.11 g
B) 18.54 g
C) 10.61 g
D) 29.75 g
E) 5.06 g
A) 10.11 g
B) 18.54 g
C) 10.61 g
D) 29.75 g
E) 5.06 g
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5
The isotope has a half-life of years. Given an initial amount of 16 grams of the isotope, how many grams will remain after 1,000 years? After 10,000 years? Round your answers to four decimal places.
A) 10.8825 g, 8.4006 g
B) 9.3278 g, 7.2005 g
C) 15.5464 g, 12.0008 g
D) 6.2185 g, 4.8003 g
E) 18.6556 g, 14.4010 g
has a half-life of years. Given an initial amount of 16 grams of the isotope, how many grams will remain after 1,000 years? After 10,000 years? Round your answers to four decimal places. A) 10.8825 g, 8.4006 g
B) 9.3278 g, 7.2005 g
C) 15.5464 g, 12.0008 g
D) 6.2185 g, 4.8003 g
E) 18.6556 g, 14.4010 g
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6
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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7
A calf that weighs 50 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) 750 lb
C) 1150 lb
D) 1250 lb
E) 1900 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) 750 lb
C) 1150 lb
D) 1250 lb
E) 1900 lb
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8
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) 1,316.250
B) 1,286.250
C) 1,236.250
D) 140.599
E) 20,580.000
, and when , . What is the value of N when ? Round your answer to three decimal places. A) 1,316.250
B) 1,286.250
C) 1,236.250
D) 140.599
E) 20,580.000
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9
Solve the differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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10
A 500-gallon tank is half full of distilled water. At time , a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 11 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 9 gallons per minute. At what time will the tank be full?
A) 250 minutes
B) 126 minutes
C) 125 minutes
D) 501 minutes
E) 1000 minutes
, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 11 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 9 gallons per minute. At what time will the tank be full? A) 250 minutes
B) 126 minutes
C) 125 minutes
D) 501 minutes
E) 1000 minutes
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11
Solve the differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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12
A 600-gallon tank is full of a solution containing 55 pounds of concentrate. Starting at time distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the amount of concentrate Q in the solution as a function of t.
A)
B)
C)
D)
E)
distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the amount of concentrate Q in the solution as a function of t.A)
B)
C)
D)
E)
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13
Find an equation of the graph that passes through the point and has the slope .
A)
B)
C)
D)
E)
and has the slope . A)
B)
C)
D)
E)
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14
Solve the differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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15
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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16
Sketch the slope field for the differential equation and use the slope field to sketch the solution satisfying the condition .
A)
B)
C)
D)
E)
and use the slope field to sketch the solution satisfying the condition . A)
B)
C)
D)
E)
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17
Solve the first order linear differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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18
The logistic function models the growth of a population. Determine when the population reaches % 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 % 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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19
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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20
Find the principal that must be invested at the rate 6%, compounded monthly, so that $3,000,000 will be available for retirement in 55 years. Round your answer to the nearest cent.
A) $909,090.91
B) $111,563.08
C) $2,280,278.32
D) $825,000.00
E) $121,702.27
A) $909,090.91
B) $111,563.08
C) $2,280,278.32
D) $825,000.00
E) $121,702.27
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21
A 200-gallon tank is full of a solution containing 25 pounds of concentrate. Starting at time distilled water is added to the tank at a rate of 20 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the time at which the amount of concentrate in the tank reaches 15 pounds. Round your answer to one decimal place.
A) 2.2 min
B) 1.8 min
C) 10.2 min
D) 5.1 min
E) 10.2 min
distilled water is added to the tank at a rate of 20 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the time at which the amount of concentrate in the tank reaches 15 pounds. Round your answer to one decimal place.A) 2.2 min
B) 1.8 min
C) 10.2 min
D) 5.1 min
E) 10.2 min
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22
Sketch the slope field for the differential equation and use the slope field to sketch the solution that passes through the point .
A)
B)
C)
D)
E)
and use the slope field to sketch the solution that passes through the point . A)
B)
C)
D)
E)
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23
Find the particular solution of the differential equation that satisfies the boundary condition .
A)
B)
C)
D)
E)
that satisfies the boundary condition . A)
B)
C)
D)
E)
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24
Find the orthogonal trajectories of the family .
A)
B)
C)
D)
E)
.
A)
B)
C)
D)
E)
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25
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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26
Find the particular solution of the differential equation that satisfies the boundary condition .
A)
B)
C)
D)
E)
that satisfies the boundary condition . A)
B)
C)
D)
E)
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k this deck
27
Select from the choices below the slope field for the differential equation.
A)
B)
C)
D)
E) none of the above
A)
B)
C)
D)
E) none of the above
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28
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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29
Identify the graph of the logistic function .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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30
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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31
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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32
Suppose that the population (in millions) of a Uganda in 2007 is 30.3 and that expected continuous annual rate of change of the population is 0.036. The exponential growth model for the population by letting corresponds to 2000 is . Use the model to predict the population of the country in 2014. Round your answer to two decimal places.
A) 32.56 millions
B) 37.61 millions
C) 40.41 millions
D) 38.98 millions
E) 31.41 millions
corresponds to 2000 is . Use the model to predict the population of the country in 2014. Round your answer to two decimal places. A) 32.56 millions
B) 37.61 millions
C) 40.41 millions
D) 38.98 millions
E) 31.41 millions
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33
The isotope has a half-life of 1,599 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) 4076.8643 g, 1713.1817 g
B) 5299.9236 g, 2227.1362 g
C) 2446.1186 g, 1027.9090 g
D) 3261.4914 g, 1370.5454 g
E) 2038.4321 g, 856.5909 g
has a half-life of 1,599 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) 4076.8643 g, 1713.1817 g
B) 5299.9236 g, 2227.1362 g
C) 2446.1186 g, 1027.9090 g
D) 3261.4914 g, 1370.5454 g
E) 2038.4321 g, 856.5909 g
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34
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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35
The logistic function models the growth of a population. Identify the initial population.
A) 6
B) 8
C) 3
D) 24
E) 2
models the growth of a population. Identify the initial population.A) 6
B) 8
C) 3
D) 24
E) 2
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36
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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37
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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38
The logistic function models the growth of a population. Identify the value of k.
A) 1.7
B) 2.2
C) 0.2
D) 20
E) 0.5
models the growth of a population. Identify the value of k.A) 1.7
B) 2.2
C) 0.2
D) 20
E) 0.5
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39
Solve the differential equation .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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40
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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41
Solve the Bernoulli differential equation .
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
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42
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)
Unlock Deck
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43
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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44
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
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45
The initial investment in a savings account in which interest is compounded continuously is $803. If the time required to double the amount is years, what is the annual rate? Round your answer to two decimal places.
A) 7.30 %
B) 7.70 %
C) 13.71 %
D) 6.10 %
E) 8.70 %
years, what is the annual rate? Round your answer to two decimal places. A) 7.30 %
B) 7.70 %
C) 13.71 %
D) 6.10 %
E) 8.70 %
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46
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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k this deck
47
Assume an object weighing 7 pounds is dropped from a height of 9,000 feet, where the air resistance is proportional to the velocity. Round numerical answers in your answer to two places.
(i) Write the velocity as a function of time if the object's velocity after 4 seconds is 2.33 feet per second.
(ii) What is the limiting value of the velocity function?
A) (i) ; (ii) 0
B) (i) ; (ii) 0
C) (i) ; (ii) 2.33
D) (i) ; (ii) 2.33
E) (i) ; (ii) limit does not exist
(i) Write the velocity as a function of time if the object's velocity after 4 seconds is 2.33 feet per second.
(ii) What is the limiting value of the velocity function?
A) (i)
; (ii) 0B) (i)
; (ii) 0C) (i)
; (ii) 2.33D) (i)
; (ii) 2.33E) (i)
; (ii) limit does not exist Unlock Deck
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48
Find an equation of the graph that passes through the point and has the slope .
A)
B)
C)
D)
E)
and has the slope . A)
B)
C)
D)
E)
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49
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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50
Use Euler's Method to make a table of values for the approximate solution of the following differential equation with specified initial value. Use 5 steps of size 0.05. ,
A)
0)000
1)000
2)000
3)000
4)000
5)000
B)
0)000
1)000
2)000
3)000
4)000
5)000
C)
0)000
1)000
2)000
3)000
4)000
5)000
D)
0)000
1)000
2)000
3)000
4)000
5)000
E)
0)000
1)000
2)000
3)000
4)000
5)000
, A)
0)000
1)000
2)000
3)000
4)000
5)000
B)
0)000
1)000
2)000
3)000
4)000
5)000
C)
0)000
1)000
2)000
3)000
4)000
5)000
D)
0)000
1)000
2)000
3)000
4)000
5)000
E)
0)000
1)000
2)000
3)000
4)000
5)000
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51
Find the logistic equation that satisfies the following differential equation and initial condition. ,
A)
B)
C)
D)
E) none of these
, A)
B)
C)
D)
E) none of these
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52
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. If the animal is sold when its weight reaches 900 pounds, find the time of sale using the model . Round your answer to two decimal places.
A) 7.02 years
B) 2.76 years
C) 1.12 years
D) 1.08 years
E) 0.44 year
where w is weight in pounds and t is time in years. If the animal is sold when its weight reaches 900 pounds, find the time of sale using the model . Round your answer to two decimal places. A) 7.02 years
B) 2.76 years
C) 1.12 years
D) 1.08 years
E) 0.44 year
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53
Solve the differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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54
A conservation organization releases 20 wolves into a preserve. After 2 years, there are 35 wolves in the preserve. The preserve has a carrying capacity of 125. Determine the time it takes for the population to reach 80.
A) 9.020 years
B) 4.875 years
C) 6.259 years
D) 3.692 years
E) 7.884 years
A) 9.020 years
B) 4.875 years
C) 6.259 years
D) 3.692 years
E) 7.884 years
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55
Use the differential equation and its slope field to find the slope at the point .
A) -8
B) -1
C) -4
D) -16
E) 8
and its slope field to find the slope at the point . A) -8
B) -1
C) -4
D) -16
E) 8
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56
Select from the choices below the slope field for the differential equation.
A)
B)
C)
D) none of the above
A)
B)
C)
D) none of the above
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57
Find the particular solution of the differential equation that satisfies the boundary condition .
A)
B)
C)
D)
E)
that satisfies the boundary condition . A)
B)
C)
D)
E)
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58
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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59
A 300-gallon tank is half full of distilled water. At time , a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 6 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 4 gallons per minute. At the time the tank is full, how many pounds of concentrate will it contain? Round your answer to two decimal places.
A) lbs
B) lbs
C) lbs
D) lbs
E) lbs
, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 6 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 4 gallons per minute. At the time the tank is full, how many pounds of concentrate will it contain? Round your answer to two decimal places.A)
lbsB)
lbsC)
lbsD)
lbsE)
lbs Unlock Deck
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60
Solve the first order linear differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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61
Find the orthogonal trajectories of the family .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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62
The number of bacteria in a culture is increasing according to the law of exponential growth. After 2 hours there are 135 bacteria in the culture and after 4 hours there are 390 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 8 hours.
(iv) After how many hours will the bacteria count be 25,000?
A) (i) 46.7341; (ii) ; (iii) 4,566.8441; (iv) 14.1787 hr
B) (i) 48.8841; (ii) ; (iii) 5,941.5613; (iv) 16.4067 hr
C) (i) 46.7341; (ii) ; (iii) 3,254.11; (iv) 11.8442 hr
D) (i) 52.5141; (ii) ; (iii) 8,693.0147; (iv) 18.5179hr
E) (i) 54.0741; (ii) ; (iii) 11,345.4782; (iv) 20.2973 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 8 hours.
(iv) After how many hours will the bacteria count be 25,000?
A) (i) 46.7341; (ii)
; (iii) 4,566.8441; (iv) 14.1787 hrB) (i) 48.8841; (ii)
; (iii) 5,941.5613; (iv) 16.4067 hrC) (i) 46.7341; (ii)
; (iii) 3,254.11; (iv) 11.8442 hrD) (i) 52.5141; (ii)
; (iii) 8,693.0147; (iv) 18.5179hrE) (i) 54.0741; (ii)
; (iii) 11,345.4782; (iv) 20.2973 hr Unlock Deck
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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
Find the general solution of the differential equation .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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65
A 300-gallon tank is full of a solution containing 55 pounds of concentrate. Starting at time distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the quantity of the concentrate in the solution as .
A) 30
B) 56
C) 55
D) 0
E) 1
distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the quantity of the concentrate in the solution as . A) 30
B) 56
C) 55
D) 0
E) 1
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66
The initial investment in a savings account in which interest is compounded continuously is $768. If the time required to double the amount is years, what is the amount after 13 years? Round your answer to the nearest cent.
A) $2,090.17
B) $1,982.88
C) $2,101.89
D) $1,582.88
E) $10,525.90
years, what is the amount after 13 years? Round your answer to the nearest cent. A) $2,090.17
B) $1,982.88
C) $2,101.89
D) $1,582.88
E) $10,525.90
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67
The isotope has a half-life of 5,715 years. After 2,000 years, a sample of the isotope is reduced to 2.1 grams. What was the initial size of the sample (in grams)? How much will remain after 20,000 years (i.e., after another 18,000 years)? Round your answers to four decimal places.
A) 1.8735 g, 0.1656 g
B) 4.2824 g, 0.3786 g
C) 2.6765 g, 0.2366 g
D) 3.7471 g, 0.3313 g
E) 3.4794 g, 0.3076 g
has a half-life of 5,715 years. After 2,000 years, a sample of the isotope is reduced to 2.1 grams. What was the initial size of the sample (in grams)? How much will remain after 20,000 years (i.e., after another 18,000 years)? Round your answers to four decimal places. A) 1.8735 g, 0.1656 g
B) 4.2824 g, 0.3786 g
C) 2.6765 g, 0.2366 g
D) 3.7471 g, 0.3313 g
E) 3.4794 g, 0.3076 g
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68
The logistic function models the growth of a population. Identify the maximum carrying capacity.
A) 10
B) 3.5
C) 3
D) 2
E) 4
models the growth of a population. Identify the maximum carrying capacity.A) 10
B) 3.5
C) 3
D) 2
E) 4
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69
A calf that weighs 70 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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70
A conservation organization releases into a preserve. After years, there are in the preserve. The preserve has a carrying capacity of . Write a logistic function that models the population of in the preserve.
A)
B)
C)
D)
E)
into a preserve. After years, there are in the preserve. The preserve has a carrying capacity of . Write a logistic function that models the population of in the preserve.A)
B)
C)
D)
E)
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71
Select from the choices below the slope field for the differential equation.
A)
B)
C)
D)
E) none of the above
A)
B)
C)
D)
E) none of the above
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72
Use as a integrating factor to find the general solution of the differential equation .
A)
B)
C)
D)
E)
as a integrating factor to find the general solution of the differential equation . A)
B)
C)
D)
E)
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73
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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74
Find the general solution of the differential equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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75
The half-life of the carbon isotope C-14 is approximately 5,715 years. If the amount left after 4,000 years is 1.3 g, what is the amount after 8,000 years? Round your answer to three decimal places.
A) 0.800 g
B) 0.628 g
C) 1.300 g
D) 0.303 g
E) 1.601 g
A) 0.800 g
B) 0.628 g
C) 1.300 g
D) 0.303 g
E) 1.601 g
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76
A phase trajectory is shown for populations of rabbits and foxes. Describe how each population changes as time goes by.
Select the correct statement.
A) At the population of foxes reaches a minimum of about 30.
B) At the number of rabbits rebounds to 500.
C) At the number of foxes reaches a maximum of about 2400.
Select the correct statement.
A) At
the population of foxes reaches a minimum of about 30.B) At
the number of rabbits rebounds to 500.C) At
the number of foxes reaches a maximum of about 2400. Unlock Deck
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77
Write and solve the differential equation that models the following verbal statement:
The rate of change of with respect to is proportional to .
A) ,
B) ,
C) ,
D) ,
E) ,
The rate of change of
with respect to is proportional to .
A)
, B)
, C)
, D)
, E)
, Unlock Deck
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78
Find the orthogonal trajectories of the family .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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79
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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80
Suppose an eight-pound object is dropped from a height of 5,000 feet, where the air resistance is proportional to the velocity. Write the velocity as a function of time if its velocity after 4 seconds is approximately -50 feet per second. Use a graphing utility or a computer algebra system. Round numerical answers in your answer to four places.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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