The population of a city is growing according to the function P(t)=50e^0.015t, where t represents year after 2000 and P(t) is in thousands of persons.

Question

The population of a city is growing according to the function P(t)=50e^0.015t, where t represents year after 2000 and P(t) is in thousands of persons. (a) Find the population in 2033. (b) solve 50e*0.015t =150

Answer 1

P(t) = 50e^0.015t

a.) Population in 2033

t = 2033 - 2000 = 33

this is because t is years since 2000

50e^0.015(33) = 50e^0.495 = 50 * 1.6405 = 82.025

82.025 * 1000 = 82,025 people in 2033

Multiply by 1000 because P(t) is in thousands of people

b.)

50e^0.015t = 150

divide both sides by 50

e^0.015t = 3

take the natural log of both sides to cancel out the e

ln(e^0.015t) = ln(3)

0.015t = ln(3)

divide both sides by 0.015

t = 73.24 years after 2000 the population reaches 150,000 people

or

in 2074 the population will exceed 150,000 people

Answer 2

part a

P(t)=50e^0.015t

P(t)=50e^0.015(33)

P(t)=1674.9 1674 people

part b

50e*0.015t =150

50e*0.015t/50 =150/50

ln(e*0.015t =3)

0.015t=ln(3)

0.015t/0.015=ln(3)/0.015

t=ln(3)/0.015

 t=73.24