Question

The population of rabbits on an island is growing exponentially. In the year 1991, the population of rabbits was 9100, and by 1998 the population had grown to 18000. Predict the population of rabbits in the year 2006, to the nearest whole number.

Answer 1

To predict the population of rabbits in the year 2006, we can use the exponential growth formula.

Step-by-step explanation:

To predict the population of rabbits in the year 2006, we can use the exponential growth formula: P(t) = P0 * e^(rt), where P(t) is the population at time t, P0 is the initial population, r is the growth rate, and e is the base of natural logarithms. We are given that the population in 1991 was 9100, so P0 = 9100. We can solve for the growth rate (r) by using the formula: 18000 = 9100 * e^(r * 7), since the population doubled in 7 years (from 1991 to 1998). Solving for r, we find that r ≈ 0.0975. Now we can use this growth rate to predict the population in 2006, which is 15 years from 1991. Substituting the values, we get: P(15) = 9100 * e^(0.0975 * 15). Evaluating this expression, we find that the population in 2006 is approximately 37,428 rabbits.

Answer 2

`39,223\ rabbits`

Step-by-step explanation:

we know that

The equation of a exponential growth function is given by

`y=a(1+r)^x`

where

y is the population of rabbits

x is the number of years since 1991

a is the initial value

r is the rate of change

we have

`a=9,100`

substitute

`y=9,100(1+r)^x`

For the year 1998

the number of years is equal to

x=1998-1991=7 years

so

we have the ordered pair (7,18,000)

substitute in the exponential equation and solve for r

`18,000=9,100(1+r)^7`

`(18,000/9,100)=(1+r)^7`

elevated both sides to 1/7

`(1+r)=1.1023`

`r=0.1023`

therefore

`y=9,100(1+0.1023)^x`

`y=9,100(1.1023)^x`

Predict the population of rabbits in the year 2006

Find the value of x

x=2006-1991=15 years

substitute the value of x in the equation

`y=9,100(1.1023)^(15)`

`y=39,223\ rabbits`