Word problem equation

Question

Suppose the population of Detroit was 1.5 million in 1990 and 1.0 million in 2002. If the city is losing population exponentially, how many years after 2002 will its population be 0.5 million?

Answer 1

1.0 = 1.5e^12k

0.667 ≈ e^12k

ln 0.667 ≈ 12k

-0.405 ≈ 12k

-0.034 ≈ k

0.5 = 1.0e^-0.034

0.5 = e^-0.034t

ln 0.5 = -0.034t

-0.693 ≈ -0.043t

16.12 ≈ t

Answer 2

The decay rate is ln(3/2)/12

Then, for the population to decline by 1/2, it will take 12 ln(2)/ln(3/2) = 20.5141354962175 years.