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grandymaker [24]
3 years ago
5

1. The number of rabbits on an island is increasing exponentially.

Mathematics
1 answer:
Zina [86]3 years ago
5 0

Answer:

<em>There are approximately 114 rabbits in the year 10</em>

Step-by-step explanation:

<u>Exponential Growth </u>

The natural growth of some magnitudes can be modeled by the equation:

P=P_o(1+r)^t

Where P is the actual amount of the magnitude, Po is its initial amount, r is the growth rate and t is the time.

We are given two measurements of the population of rabbits on an island.

In year 1, there are 50 rabbits. This is the point (1,50)

In year 5, there are 72 rabbits. This is the point (5,72)

Substituting in the general model, we have:

50=P_o(1+r)^1

50=P_o(1+r)\qquad\qquad[1]

72=P_o(1+r)^5\qquad\qquad[2]

Dividing [2] by [1]:

\displaystyle \frac{72}{50}=(1+r)^{5-1}=(1+r)^{4}

Solving for r:

\displaystyle r=\sqrt[4]{\frac{72}{50}}-1

Calculating:

r=0.095445

From [1], solve for Po:

\displaystyle P_o=\frac{72}{(1+r)^5}

\displaystyle P_o=\frac{72}{(1.095445)^5}

P_o=45.64355

The model can be written now as:

P=45.64355(1.095445)^t

In year t=10, the population of rabbits is:

P=45.64355(1.095445)^{10}

P = 113.6

P\approx 114

There are approximately 114 rabbits in the year 10

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