The population of rabbits on an island is growing exponentially. In the year 2004, the population of rabbits was 820, and by 2009 the population had grown to 1370. Predict the population of rabbits in the year 2013, to the nearest whole number.

When a species is introduced to a new environment, it often has no

natural predators and multiplies quickly. This situation occurred in

Australia in 1859, when a landowner named Thomas Austin released

24 rabbits for hunting. The rabbits reproduced so quickly that within

20 years they were referred to as a “grey carpet” on the continent,

and drove many native plant and animal species to extinction.

Because a pair of rabbits can produce an average of 7 surviving baby

rabbits a year when in a dense environment, you can estimate that

the rabbit population multiplied by a factor of __7

2

, or 3.5, each year.

To model this situation, let r0 = 24 be the initial population of rabbits

in 1859. This is similar to using h0 for initial height and v0 for initial

velocity in previous formulas. Then let rn = the number of rabbits

n years after 1859. A recursive formula for a sequence modeling this

situation is

{ r0 = 24 rn = 3.5rn-1, for n≥1

.

This is a geometric sequence with starting term 24 and constant ratio

3.5. The table below shows the population rn predicted by the model

for n = 1, 2, 3, … , 10 years after 1859, rounded to the nearest whole

number of rabbits. The fi rst four ordered pairs are graphed below.An explicit formula for this sequence is rn = 24(3.5)n for n ≥ 0.

By representing the population as the function f with equation

r = f(t) = 24(3.5)t

, you can estimate the population r at any real

number of years t ≥ 0.

QY1

In the graph at the left below, r = 24(3.5)t

is plotted for values of t

from 0 to 3.75, increasing by 0.25. The middle graph shows values for

t from 0 to 3.8, increasing by 0.02. Number of Rabbits

200

600

1000

1400

1800

2200

2600

0

Years after 1859

2 4 6 8 10

t

r

Number of Rabbits

200

600

1000

1400

1800

2200

2600

0

Years after 1859

2 4 6 8 10

t

r

t

r

Number of Rabbits

200

600

1000

1400

1800

2200

2600

Years after 1859

0 2 4 6 8 10

Because time is continuous when measuring population growth, you

can think of the function with equation r = f(t) = 24(3.5)t

as being

defi ned for all real nonnegative values of t, as graphed above at the

right. However, the equation has meaning for any real number t.

Using the set of real numbers as the domain results in the function

graphed at the right below.

This graph shows an exponential curve. The

shape of an exponential curve is different from

the shape of a parabola, a hyperbola, or an arc of a

circle. The range of the function f is the set of

positive real numbers. Its graph never intersects

the t-axis, but gets closer and closer to it as t gets

smaller and smaller. Thus, the t-axis is a

horizontal asymptote to the graph. Substituting

t = 0 into the equation gives an r-intercept of 24.

This represents the number of rabbits present

when they were fi rst introduced.

answer: 5100 just subtract

step-by-step explanation: