Question

1. during a certain period of time, about 70 northern sea otters had an annual growth of 18%. use an exponential function to find the number of sea otters after 4 years.
2. a population of fish starts at 8,000 and decreases by 6% per year. use an exponential function to find the population of fish in 10 years.
3. twenty years ago, mr. davis purchased his home for $160,000. since then, the value of the home has increased about 5% per year. use an exponential function to find the value of the home today.

© gina wilson (all things algebra®, llc), 2012 - 2017

Explanation:

Problem 6: Sea Otter Population

Step1: Define growth function

Exponential growth formula: $P(t) = P_0(1+r)^t$
Where $P_0=70$, $r=0.18$, $t=4$

Step2: Substitute values

$P(4) = 70(1+0.18)^4$

Step3: Calculate growth factor

$1.18^4 \approx 1.9388$

Step4: Compute final population

$P(4) = 70 \times 1.9388$

Problem 7: Fish Population

Step1: Define decay function

Exponential decay formula: $P(t) = P_0(1-r)^t$
Where $P_0=8000$, $r=0.06$, $t=10$

Step2: Substitute values

$P(10) = 8000(1-0.06)^{10}$

Step3: Calculate decay factor

$0.94^{10} \approx 0.5386$

Step4: Compute final population

$P(10) = 8000 \times 0.5386$

Problem 8: Home Value

Step1: Define growth function

Exponential growth formula: $V(t) = V_0(1+r)^t$
Where $V_0=160000$, $r=0.05$, $t=20$

Step2: Substitute values

$V(20) = 160000(1+0.05)^{20}$

Step3: Calculate growth factor

$1.05^{20} \approx 2.6533$

Step4: Compute final value

$V(20) = 160000 \times 2.6533$

Answer:

1. Number of sea otters after 4 years: $\approx 136$
2. Population of fish after 10 years: $\approx 4309$
3. Value of the home today: $\approx \$424,528$