Question

The population of a town is growing according to the differential equation dy/dt = ky The growth constant, k, is equal to 0.13 year^-1. The size of the population at the start of the year 2000 was 6 thousand. Since the population is growing exponentially, the population in year t is given by y = 6e^0.13t Here, y is measured in thousands and t is measured in years since 2000. a. What is the population of the town at the start of the year 2007? (Enter your answer correct to one decimal place.) Population: thousand. b. How many years does it take for the population to double? (Enter your answer correct to two decimal places.) Doubling time : years. c. What will be the size of the population after three doubling times have passed? (That is, after three times the correct answer to part b have passed?) (Enter your answer correct to one decimal place.) Population after three doubling times : thousand.

          The population of a town is growing according to the differential equation dy/dt = ky
The growth constant, k, is equal to 0.13 year^-1.
The size of the population at the start of the year 2000 was 6 thousand.
Since the population is growing exponentially, the population in year t is given by y = 6e^0.13t
Here, y is measured in thousands and t is measured in years since 2000.
a. What is the population of the town at the start of the year 2007?
(Enter your answer correct to one decimal place.)
Population: thousand.
b. How many years does it take for the population to double?
(Enter your answer correct to two decimal places.)
Doubling time : years.
c. What will be the size of the population after three doubling times have passed? (That is, after three times the correct answer to part b have passed?)
(Enter your answer correct to one decimal place.)
Population after three doubling times : thousand.

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The population of a town is growing according to the differential equation dy/dt = ky
The growth constant, k, is equal to 0.13 year^-1.
The size of the population at the start of the year 2000 was 6 thousand.
Since the population is growing exponentially, the population in year t is given by y = 6e^0.13t
Here, y is measured in thousands and t is measured in years since 2000.
a. What is the population of the town at the start of the year 2007?
(Enter your answer correct to one decimal place.)
Population: thousand.
b. How many years does it take for the population to double?
(Enter your answer correct to two decimal places.)
Doubling time : years.
c. What will be the size of the population after three doubling times have passed? (That is, after three times the correct answer to part b have passed?)
(Enter your answer correct to one decimal place.)
Population after three doubling times : thousand.

Added by David C.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert William Semus

10/08/2022

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13t), where t = 7 years since 2000. Show more…

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The population of a town is growing according to the differential equation dy/dt = ky. The growth constant, k, is equal to 0.13 year^-1. The size of the population at the start of the year 2000 was 6 thousand. Since the population is growing exponentially, the population in year t is given by y = 6e^0.13t. Here, y is measured in thousands and t is measured in years since 2000. a. What is the population of the town at the start of the year 2007? (Enter your answer correct to one decimal place.) Population: thousand. b. How many years does it take for the population to double? (Enter your answer correct to two decimal places.) Doubling time: years. c. What will be the size of the population after three doubling times have passed? (That is, after three times the correct answer to part b have passed?) (Enter your answer correct to one decimal place.) Population after three doubling times: thousand.

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00:01 So in this question, we say the population of a town is growing according to a differential equation, dy, d .d, t, equals ky.

00:08 The growth constant k is equal to 0 .07 per year.

00:13 The size of the population at the start of the year 2000 was 15 ,000.

00:18 Since the population is growing exponentially, they say the population in e or t is given by y equals 15e to the power of 0 .000.

00:30 0 .07 t.

00:32 Here the y is measured in thousands and t is measured in years since 2000.

00:39 In part a, they say what is the population of the town at the start of the year 2006? so 2006 is going to be t equals six and so what i'm going to do is just plug that in to this equation.

01:00 I have 15 to the power of .07 times six.

01:07 15e to the power of 0 .07 times six.

01:12 Now you get 22 .829.

01:16 Now, they did say i want the population.

01:19 This was in thousands.

01:22 So this is going to give you 22 ,829 .42.

01:33 That would be my population in the year 2006.

01:38 So we let t equals six.

01:41 In part b, what is the population of the town at the end of the year 2012? you've got to be careful.

01:49 So the end of 2012 is the same as the very beginning of 2013.

01:55 So i'm going to be plugging not 12, but 13, into this equation.

02:00 Getting y equals 15 e to the power of 0 .07 times 13.

02:10 And so i'm going to have the exact same setup, except my t is 13 instead of 6.

02:18 And now i'm getting 37 ,264 .8.

02:24 I'm getting 37 ,000, 37 ,264...

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