Question

answer the following questions.

1. choose the correct answers.

a herd of buffalo began with a population of 75,000. the average birth rate has been 28% per year, but the death rate has been 43% per year.
what is the difference between the birth rate and death rate?
is this population in a state of growth or decay?
write an exponential equation modeling the population’s change over time.
according to your model, what will the population of buffalo in this herd be in 10 years?
a. 71%
b. decay
c. 14,765
d. growth
e. $a(t) = 75,000(0.85)^t$
f. $a(t) = 75,000(1.15)^t$
g. 63,750
h. 15%

Explanation:

Step1: Calculate rate difference

Subtract death rate from birth rate:
$28\% - 43\% = -15\%$
The magnitude of the difference is $15\%$.

Step2: Determine growth/decay

Since net rate is negative ($-15\%$), the population shrinks, so it is decay.

Step3: Build exponential model

Exponential decay formula: $A(t) = A_0(1+r)^t$, where $A_0=75000$, $r=-0.15$.
$1 + r = 1 - 0.15 = 0.85$
So $A(t) = 75000(0.85)^t$

Step4: Calculate population at 10 years

Substitute $t=10$ into the model:
$A(10) = 75000(0.85)^{10}$
First calculate $(0.85)^{10} \approx 0.19687$
Then $75000 \times 0.19687 \approx 14765$

Answer:

1. What is the difference between the birth rate and death rate? h. 15%
2. Is this population in a state of growth or decay? b. decay
3. Write an exponential equation modeling the population's change over time. e. $A(t) = 75,000(0.85)^t$
4. According to your model, what will the population of buffalo in this herd be in 10 years? c. 14,765