Question

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

Answer 1

The goat population reaches 1000 in 12.4 years

Explanation:

After t years, the number of goats is given by

`N = N_0e^(bt)`

where
`N_0` is the initial number of goats and b is the per capita growth rate.

From the question,

• 
  `N_0 = 2`
• b = 0.5
• N = 1000

`1000 = 2e^(0.5t)`

`e^(0.5t) = 500`

`0.5t = \ln 500 = 6.214`

`t = (6.2)/(0.5) = 12.4`

Answer 2

12.4 years

Explanation:

Let recall the following,

The number of goats after t years is shown below

N = N₀∧ebt

Let N₀ = the number of the initial goats

b = The capital rate of growth

From the example given,

Let insert the values

N₀ is = 2

b = 0.5

Let N = 1000

So,

1000= 2e∧0.5t

e∧0.5t = 500

0.5t = ln 500 = 6.214

Therefore

t = 6.214/0.5 =12.4

Therefore, the number of years it will take for the goat population to reach 1000 is = 12.4 years