Question

Question 4, In 2011, the population of a colony is 17000, and is decreasing exponentially at a rate of 6.6% per year.

(a) What will the population be in the year 2033?

(b) In what year will half of the population be left?

Answer 1

Initial value : 17000

As it is exponential decay,

y = 17000e^-kt

k : decay rate expressed in decimal

k = 0.066

y = 17000e^-0.066t

t : number of years since 2011

(a) t = 22 as the year is 2033 (22 years since 2011)

y = 17000e^-0.066(22)

y = 17000e^-1.452

y = 3979.73 ≈ 3980

Population in 2033 = 3980

(b) y = 17000/2 = 8500

8500 = 17000e^-0.066t

1/2 = e^-0.066t

-0.066t = ln(1/2)

t = ln(1/2) / -0.066

t = 10.5 years

After 10.5 years i.e. in 2011 + 10.5 = 2021.5.

2022