g A sailing ship stopped at several islands, releasing one mating pair of goats onto each one. On the large island, the goats' g

Question

2 months ago

14

g A sailing ship stopped at several islands, releasing one mating pair of goats onto each one. On the large island, the goats' g

rowth rate was approximately exponential for a long time. If their per capita growth rate is 0.5 per year, how many years will it take before the goat population reaches 1000

Mathematics

2 answers:

tester [12.3K] · Answer 1 · 2026-03-07T13:56:33+03:00

tester [12.3K]2 months ago

5 0

Response:

The goat population will reach 1000 in a time span of 12.4 years

Detailed explanation:

After t years, the population of goats is described by

$N = N_0e^{bt}$

where $N_0$ indicates the initial count of goats and b is the growth rate per capita.

Based on the problem,

$N_0 = 2$
b = 0.5
N = 1000

$1000 = 2e^{0.5t}$

$e^{0.5t} = 500$

$0.5t = \ln 500 = 6.214$

$t = \dfrac{6.2}{0.5} = 12.4$

tester [12.3K] · Answer 2 · 2026-03-07T13:54:15+03:00

Response:

12.4 years

Detailed reasoning:

Let’s revisit the following:

The total goat population after t years is represented as follows

N = N₀∧ebt

Let N₀ equal the initial goat count

b represents the growth capital rate

In the current example,

We will substitute the values:

N₀ = 2

b = 0.5

Let N be 1000

Thus,

1000 = 2e∧0.5t

e∧0.5t = 500

0.5t = ln(500) = 6.214

Hence

t = 6.214 / 0.5 = 12.4

Accordingly, it will take 12.4 years for the goat population to reach 1000.