Chapter 4.1, Problem 42PS

Solution

(A)

To find: Write a model giving the number $n$ of breeding pairs of bald eagle in $t$ years.

The a model giving the number $n$ of breeding pairs of bald eagle in $t$ years is $n=41\left(1.089\right){}^t$ .

Given information:

Given that there were $41$ breeding pairs over the next 24 years, the number of breeding pairs increased by about $8.9\%$ each year.

Calculation:

Exponential growth model will be used here which is $p=a{\left(1+r\right)}^t$ here, $a$ is initial value $1+r$ is is the growth factor.

Given $a=41years$ , $r=\frac{8.9}{100}$ .

Substitute the given values in $p=a\left(1+r\right){}^t$ :

  $\begin{array}{l}n=41{\left(1+.089\right)}^t\\ n=41\left(1.089\right){}^t\end{array}$

(B)

To find: Write table values for model.

The table is given.

Given information:

Given that there were $41$ breeding pairs over the next 24 years, the number of breeding pairs increased by about $8.9\%$ each year.

Calculation:

The table for the population after $t$ years will be calculated by substituting $t=1,2,\mathrm{3......24}$ in

  $n=41\left(1.089\right){}^t$

  $\begin{array}{l}n=41\left(1.089\right){}^1\\ n=44.689\end{array}$

Similarly, other values can be found

The table is given below:

Time (t)	Population
1	44.689
2	48.62
3	52.92
4	57.66
5	62.79
6	68.38
7	74.47
8	81.10
9	88.32
10	96.18
11	104.73
12	114.06
13	124.21
14	135.26
15	147.30
16	160.41
17	174.69
18	190.23
19	207.16
20	225.60
21	245.68
22	267.88
23	291.36
24	317.29

(c)

To find: Draw the graph of the model

Given information:

Given that there were $41$ breeding pairs over the next 24 years, the number of breeding pairs increased by about $8.9\%$ each year.

Calculation:

Plot the point which are found in the table in part $\left(B\right)$ .

The graph of the model will be:

  

(c)

To find: Find from the graph How many breeding pair of bald eagle in 2001.

It can be observe from the graph that the number of breeding pair will be 350.

Given information:

Given that there were $41$ breeding pairs over the next 24 years, the number of breeding pairs increased by about $8.9\%$ each year

  

Calculation:

Starting from $0=1$ for 1977 for year 2001 $t=24$

It can be observe from the graph that the number of breeding pair will be 350.