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Question : Sales (in thousands) of a certain product are declining at a rate proportional : 2151808

Solve the problem.

31) Sales (in thousands) of a certain product are declining at a rate proportional to the amount of sales, with a decay constant of 6% per year. Write a differential equation to express the rate of sales decline.

A) dy/dt = -0.06t

B) dy/dt = -0.06y

C) dy/dt = -0.94y

D) dy/dt = e^{-0.06t}

32) The amount of a radioactive substance decreases exponentially, with a decay constant of 4% per month. Find a general solution to the differential equation which expresses the rate of change.

A) y = -Me^{0.04t}

B) y = -0.04t

C) y = Me^{-0.96t}

D) y = Me^{-0.04t}

33) Sales (in thousands) of a certain product are declining at a rate proportional to the amount of sales, with a decay constant of 8% per year. By writing and solving a differential equation, determine how much time will pass before sales become 60% of their original value.

A) 7.3 years

B) 5.7 years

C) 6.8 years

D) 6.4 years

34) The population of a country is predicted to increase from 21.5 million in 2000 to 49.5 million in 2050. Assuming the unlimited growth model dy/dt = ky fits this population growth, express the population y as a function of the year t. Let 2000 correspond to t = 0.

A) y = 21.5e^{0.01468t}

B) y = 21.5e^{0.01668t}

C) y = 49.5e^{0.01668t}

D) y = (49.5/1 + e^{-0.01468t})

35) Assume that the rate of change of population of a certain city is given by (dy/dt) = 6000e^{0}.06t, where y is the population at time t, in years. The population was 100,000 in 1980 (t = 0 in 1980). Predict the population in 1990.

A) 202,212

B) 186,212

C) 192,212

D) 182,212

36) A wild animal preserve can support no more than 200 elephants. 35 elephants were known to be in the preserve in 1980. Assume that the rate of growth of the population is

(dP/dt) = 0.0003P(200 - P)

where t is time in years. Find a formula for the elephant population in terms of t. Let 1980 correspond to t = 0.

A) P = (200/1 - 4.71e^{-0.6t})

B) P = (200/1 + 4.71e^{-0.6t})

C) P = (200/1 + 5.71e^{-0.06t})

D) P = (200/1 + 4.71e^{-0.06t})

37) A wild animal preserve can support no more than 150 elephants. 40 elephants were known to be in the preserve in 1980. Assume that the rate of growth of the population is

(dP/dt) = 0.0004P(150 - P)

where t is time in years. How long will it take for the elephant population to increase from 40 to 100? [First find a formula for the elephant population in terms of t.]

A) 30.7 years

B) 28.4 years

C) 26.1 years

D) 29.5 years

38) When a dead body is discovered, one of the first steps in the ensuing investigation is for a medical examiner to determine the time of death as closely as possible. If the temperature of the medium has been fairly constant and less than 48 hours have passed since death, Newton's law of cooling can be used. Newton's law of cooling states, (dT/dt) = -k(T - T with subscript((M))), where k is a constant, T is the temperature of the object after t hours, and T with subscript((M)) is the (constant) temperature of the surrounding medium. Assuming the temperature of a body at death is 98.6°F, the temperature of the surrounding air is 71°F, and at the end of one hour the body temperature is 89°F, what is the temperature of the body after 4 hours? Round to the nearest tenth of a degree.

A) 89°F

B) 76°F

C) 5°F

D) 71.5°F

39) When a dead body is discovered, one of the first steps in the ensuing investigation is for a medical examiner to determine the time of death as closely as possible. If the temperature of the medium has been fairly constant and less than 48 hours have passed since death, Newton's law of cooling can be used. Newton's law of cooling states, (dT/dt) = -k(T - T with subscript((M))), where k is a constant, T is the temperature of the object after t hours, and T with subscript((M)) is the (constant) temperature of the surrounding medium. Assuming the temperature of a body at death is 98.6°F, the temperature of the surrounding air is 67°F, and at the end of one hour the body temperature is 92°F, when will the temperature of the body be 78°F? Round to the nearest tenth of an hour.

A) 1.3 hr

B) 0.4 hr

C) 1.5 hr

D) 4.5 hr

A person's weight depends both on the daily rate of energy intake, say C calories per day, and the daily rate of energy consumption, typically between 12 and 20 calories per pound per day. Using an average value of 16 calories per pound per day, a person weighing w pounds uses 16w calories per day. If C = 16w, then weight remains constant, and weight gain or loss occurs according to whether C is greater or less than 16w.

To determine how fast a change in weight will occur, a plausible assumption is that dw/dt is proportional to the net excess (or deficit) C - 16w in the number of calories per day.

40) Assume C is constant and write a differential equation to express this relationship. Use k to represent the constant of proportionality.

A) (dw/dt) = k(16w - C)

B) (dw/dt) = C(k - 16w)

C) (dw/dt) = k(C - 16w)

D) (dw/dt) = k(C + 16w)

41) Assuming C is constant, a differential equation to express this relationship is dw/dt = k(C - 16w), where k is the constant of proportionality. The units of dw/dt are pounds per day, and the units of C - 16w are calories per day. What units must k have?

A) calorie/pounds

B) pounds/day

C) pounds/calorie

D) pounds calories/day^{2}

42) Write a differential equation using the fact that 3500 calories is equivalent to one pound.

A) dw/dt = 3500(C - 16w)

B) dw/dt = (16w - C)/3500

C) dw/dt = (C - 16w)/3500

D) dw/dt = C(3500 - 16w)

43) Given that 3500 calories is equivalent to one pound, a differential equation to express this relationship is dw/dt = (C - 16w)/3500. Solve this differential equation.

A) w = C - e^{-0.0046M} e^{-0.0046t }

B) w = C/16 - e^{-0.0046M} e^{-0.0046t }/16

C) w = C - e^{-0.0065M} e^{-0.0065t }

D) w = C/16 - e^{-0.0065M} e^{-0.0065t }/16

Solve the problem.

44) The table shows the population of a certain city for selected years between 1950 and 2003.

Use the logistic regression function on your calculator to determine the logistic equation that best fits the data.

A) P = (266,076.8/1 + 23.128e^{-0.1215t} )

B) P = (271,976.2/1 + 24.361e^{-0.1345t} )

C) P = (278,715.3/1 + 25.311e^{-0.1374t} )

D) P = (254,180.3/1 + 26.118e^{-0.1402t} )

45) The table shows the population of a certain city for selected years between 1950 and 2003.

By using your calculator to find the logistic regression equation that best fits the data, determine the limiting size of the population.

A) 266,078

B) 254,180

C) 271,976

D) 278, 715

46) The table shows the population of a certain city for selected years between 1950 and 2003.

By using your calculator to find the logistic regression equation that best fits the data, determine when the population of the city will first exceed 271,000.

A) In 2018

B) In 2015

C) In 2010

D) In 2012

47) The table shows the population of a certain city for selected years between 1950 and 2003.

Write a logistic differential equation in the form (dP/dt) = kP(M - P) that models the growth of the population. [You will first need to use your calculator find the logistic regression equation that best fits the data.]

A) (dP/dt) = (4.741 × 10^{-7})P(278,903.2 - P)

B) (dP/dt) = 0.1207 P(278,903.2 - P)

C) (dP/dt) = (4.945 × 10^{-7})P(271,976.2 - P)

D) (dP/dt) = 0.1345 P(271,976.2 - P)

48) The table shows the population of a certain city for selected years between 1950 and 2003.

Use the logistic regression function on your calculator to determine the logistic equation that best fits the data.

A) P = (494,193.8/1 + 24.126e^{-0.1084t} )

B) P = (482,549.6/1 + 22.095e^{-0.1132t} )

C) P = (499,107.3/1 + 23.521e^{-0.0981t} )

D) P = (478,549.6/1 + 21.095e^{-0.1209t} )

49) The table shows the population of a certain city for selected years between 1950 and 2003.

By using your calculator to find the logistic regression equation that best fits the data, determine the limiting size of the population.

A) 482,550

B) 499,107

C) 494,194

D) 478,550

50) The table shows the population of a certain city for selected years between 1950 and 2003.

By using your calculator to find the logistic regression equation that best fits the data, determine when the population of the city will first exceed 480,000.

A) In 2015

B) In 2023

C) In 2020

D) In 2018

51) The table shows the population of a certain city for selected years between 1950 and 2003.

Write a logistic differential equation in the form (dP/dt) = kP(M - P) that models the growth of the population. [You will first need to use your calculator find the logistic regression equation that best fits the data.]

A) (dP/dt) = (1.623 × 10^{-7})P(499,107.3 - P)

B) (dP/dt) = (2.193 × 10^{-7})P(494,193.8 - P)

C) (dP/dt) = (2.526 × 10^{-7})P(478,549.6 - P)

D) (dP/dt) = (2.346 × 10^{-7})P(482,549.6 - P)