Answer :
Answer:
The population function is given below as
[tex]p(t)=420(0.71)^t[/tex]The exponential formula is given blow as
[tex]\begin{gathered} p(t)=ab^t \\ \text{where,} \\ b=1+r(\text{for growth)} \\ b=1-r(\text{for decay)} \end{gathered}[/tex]Step 1:
To figure out the initial population size, we will substitute the value of t=0
[tex]\begin{gathered} p(t)=420(0.71)^t \\ p(0)=420(0.71)^0 \\ P(0)=420\times1 \\ P(0)=420 \end{gathered}[/tex]Hence,
The initial population size is = 420
Step 2:
To figure out if the function represents growth or decay, we will use the relation below
[tex]\begin{gathered} 1+r<1(\text{decay)} \\ 1+r>1(\text{growth)} \end{gathered}[/tex]The value of b in the equation is
[tex]b=0.71\text{ <1}[/tex]Therefore,
The function in the question represents DECAY
Step 3:
To figure the percentage at which each population size change each year, we will use the formula below
[tex]\%change=\frac{present\text{ population-previous population}}{\text{previous population}}\times100\%[/tex]To figure out a present population, we will substitute the value of t to be t=1
[tex]\begin{gathered} p(t)=420(0.71)^t \\ P(1)=420(0.71)^1 \\ P(1)=420\times0.71 \\ P(1)=298.2 \end{gathered}[/tex][tex]\begin{gathered} \%change=\frac{present\text{ population-previous population}}{\text{previous population}}\times100\% \\ \%change=\frac{P(0)-P(1)}{P(0)}\times100\% \\ \%change=\frac{420-298.2}{420}\times100\% \\ \%change=\frac{121.8}{420}\times100\% \\ \%change=29\% \end{gathered}[/tex]Hence,
The percentage is = 29%
