Question

if a flock of ducks is growing by 6 percent per year and starts with a population of 68 about how many ducks will be there in 10 years

Answer 1

A flock of ducks growth

We know that the next year the flock of ducks will have 6% more than the current year. If the current year the number of ducks is x, then

0.06 · x = the increase number

Then, the population of ducks next year will be

x + 0.06x = number of ducks next year

we can simplify the equation:

1.06x = number of ducks next year

Two years after, then number of ducks will be:

1.06 · number of ducks next year = number of ducks two years after

using the equation we found:

1.06 · (1.6x) = number of ducks two years after

1.06²x = number of ducks two years after

Similarly, three years after will be

1.06³x = number of ducks three years after

If we keep writing equations for every year, we will find a relation between the number of years that pass and the exponent...

n years after will be:

1.06ⁿx = number of ducks n years after

Since the current year the population is 68, then

1.06ⁿ · 68 = number of ducks n years after

We want to find the number of ducks after 10 years. This is n = 10:

`1.06^(10)\cdot68=\text{ number of ducks 10 years after}`

Since

`\begin{gathered} 1.06^(10)=1.79 \\ 1.79\cdot68\approx121.78 \end{gathered}`

Then, the equation we found says that:

number of ducks 10 years after = 121.78

But it is not possible because we cannor have 121.78 ducks, we always have an integer. Then we round it to the nearest integer: 122

Then

answer - the number of ducks 10 years after will be 122