Chapter 7.6, Problem 52PPS

(a)

To determine

The number of deer after $4\frac{1}{2}$ years if the number of deer after $\left(t\right)$ years is given by the relation $D=100\cdot 2^{\frac{t}{2}}$

Answer to Problem 52PPS

The number of deer after $4\frac{1}{2}$ years will be 475.

Explanation of Solution

Given information: The number of Deer $\left(D\right)$ after $\left(t\right)$ years is given by the relation,

  $D=100\cdot 2^{\frac{t}{2}}\mathrm{......}\left(1\right)$

Calculation: From (1) the number of deer after $4\frac{1}{2}$ years can be evaluated by plugging $\left(t=4\frac{1}{2}\right)$ in (1). Therefore from (1),

  $\begin{array}{c}D=100\cdot 2^{\frac{4.5}{2}}\\ =475.68\end{array}$

(b)

To determine

Draw a table that shows the population of deer every year for next five years.

Answer to Problem 52PPS

Time (t)	Number of Deer (D)
0	100*20 = 100
1	100*20.5 = 141
2	100*21 = 200
3	100*21.5 = 282
4	100*22 = 400
5	100*22.5 = 565

Explanation of Solution

Given information: The number of Deer $\left(D\right)$ after $\left(t\right)$ years is given by the relation,

  $D=100\cdot 2^{\frac{t}{2}}$

Calculation: From (1) the number of deer for the next five years can be evaluated by plugging $\left(t\right)=0,1,2,3,4\text{ and }5$ in (1). Therefore with the help of (1) the table is constructed as,

Time (t)	Number of Deer (D)
0	100*20 = 100
1	100*20.5 = 141
2	100*21 = 200
3	100*21.5 = 282
4	100*22 = 400
5	100*22.5 = 565

(c)

To determine

Show the graph for number of deer verses time with the help of table obtained.

Explanation of Solution

Given information: The number of Deer $\left(D\right)$ after $\left(t\right)$ years is given by the relation,

  $D=100\cdot 2^{\frac{t}{2}}$

Graph: The graph for the number of deer verses the time is shown as,

  

(d)

To determine

With the help of graph and table decide whether it is a reasonable trend over the long period. Explain.

Answer to Problem 52PPS

No, it is not reasonable trend over the long period.

Explanation of Solution

Given information: The number of Deer $\left(D\right)$ after $\left(t\right)$ years is given by the relation $D=100\cdot 2^{\frac{t}{2}}$ .

Calculation: From the table and graph of the number of deer verses the time it can be said that the number of deer grows rapidly. And as seen from the number of deer graph it can be observed that the deer grows rapidly in small interval of time and the graph line for the population of deer is approaching to y-axis. Therefore, for the long term period it is not reasonable to say anything about the trend because there are many factors like environment conditions, temperature and rate of death etc. which can affect the deer population for long term.