MATH HELP!!!!! sequences&series?

Question

In 1998 there were 3000 koalas on Koala Island. Since then, the population of koalas on the island has increase by 5% each year.

1. How many koalas were on the island in 2001?

2. In what year will the population first exceed 5000?

PLEASE EXPLAIN WITH THE WORKS SHOWN

THXXXXXXX

Answer 1

Let t = 0 be the year 1998, and P(initial) = 3000

Population function is P(t) = P(initial) * (1.05)^t

For the year 2001, t = 3 and

P(3) = 3000 * (1.05)^3 ≈ 3,473.

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To find when population exceeds 5000, set

5000 = 3000 * (1.05)^t and attempt to solve for t

(5000/3000) = (1.05)^t

(5/3) = (1.05)^t

I couldn't see the way to solve this algebraically, so I just used my calculator trying different values of t.

t=10 gives (1.05)^10 ≈ 1.63 < (5/3) so the population will not have reached 5000 yet.

t=11 gives (1.05)^11 ≈ 1.71 > (5/3) so the population exceeds 5000 in the 11th year after 1998, or the year 2009.

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Check using the population function:

P(10) = 3000 * (1.05)^10 ≈ 4,887 {Year 2008}

P(11) = 3000 * (1.05)^11 ≈ 5,131 {Year 2009}