Please help me solve this

Question

In 1998, the population of a colony is 10,000, and is decreasing exponentially at 1.5% per year.

a) What will the population be after 5 years?

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

Answer 1

Since this is decreasing exponentially, this is a decay.

Given:

r=rate = 1.5% = .015

P_o = the initial population =10,000

The formula is:

P = P_o (1- r)^t

Where P = resulting population

(a) t = time in years = 5

P = P_o (1- r)^t

P = 10,000(1-.015)⁵

P = 10,000(0.985)⁵

P ≈ 9272

(b)P =5,000

5000 =10000(1-.015)^t

1/2 = (0.985)^t

Get the natural log (ln) of both sides:

ln(1/2) = t•ln(0.985)

t = ln(1/2) / ln(0.985)

t ≈ 45.86 ≈ 46 years

1998+46 = 2044

Therefore the population will be half around the year 2044.

Answer 2

1) P(t) = P₀(1-.015)^t where t is in years. t = 5 and P₀ = 10000

2) 1/2 = (1-.015)^{t (half-life)} (take log or ln of both sides and solve for t)

Good luck!

Answer 3

a)

A = 10000(98.5)⁵

b)

0.5 = (98.5)^t