2. The trees in a forest are suffering from a disease. The population of trees, [tex]\( p \)[/tex], in thousands, is modeled by the equation [tex]\( p=600 \cdot\left(\frac{4}{5}\right)^t \)[/tex], where [tex]\( t \)[/tex] is the number of years since 2000.

a. What was the tree population in 2001? What about in 1999?

b. In the equation, what does the 600 tell us about the situation?

c. What does the number [tex]\( \frac{4}{5} \)[/tex] in the equation for [tex]\( p \)[/tex] tell you about the population?

d. Graph the equation for values of [tex]\( t \)[/tex] between 0 and 4.



Answer :

Sure, let's dive into the problem step by step.

### Part a: Tree Population in 2001 and 1999

The equation modeling the tree population is:

[tex]\[ p = 600 \cdot \left(\frac{4}{5}\right)^t \][/tex]

#### Tree Population in 2001

1. In 2001, [tex]\( t = 1 \)[/tex] since it is 1 year after 2000.
2. Substitute [tex]\( t = 1 \)[/tex] into the equation:

[tex]\[ p = 600 \cdot \left(\frac{4}{5}\right)^1 = 600 \cdot \frac{4}{5} = 480 \][/tex]

So, the tree population in 2001 was 480 thousand trees.

#### Tree Population in 1999

1. In 1999, [tex]\( t = -1 \)[/tex] since it is 1 year before 2000.
2. Substitute [tex]\( t = -1 \)[/tex] into the equation:

[tex]\[ p = 600 \cdot \left(\frac{4}{5}\right)^{-1} = 600 \cdot \frac{5}{4} = 750 \][/tex]

So, the tree population in 1999 was 750 thousand trees.

### Part b: Interpretation of 600

The coefficient 600 in the equation represents the initial tree population in the year 2000. This means that at [tex]\( t = 0 \)[/tex], the population of trees was 600 thousand.

### Part c: Interpretation of [tex]\( \frac{4}{5} \)[/tex]

The [tex]\( \frac{4}{5} \)[/tex] in the equation tells us that the population decreases each year to 80% of its size the previous year. In other words, this decay factor indicates the proportion of the population that survives from one year to the next.

### Part d: Graphing the Equation for [tex]\( t \)[/tex] Between 0 and 4

To graph the equation [tex]\( p = 600 \cdot \left(\frac{4}{5}\right)^t \)[/tex] for [tex]\( t \)[/tex] ranging from 0 to 4, let's calculate the population for each integer value of [tex]\( t \)[/tex] in this range:

- At [tex]\( t = 0 \)[/tex]:

[tex]\[ p = 600 \cdot \left(\frac{4}{5}\right)^0 = 600 \cdot 1 = 600 \][/tex]

- At [tex]\( t = 1 \)[/tex]:

[tex]\[ p = 600 \cdot \left(\frac{4}{5}\right)^1 = 480 \][/tex]

- At [tex]\( t = 2 \)[/tex]:

[tex]\[ p = 600 \cdot \left(\frac{4}{5}\right)^2 = 384 \][/tex]

- At [tex]\( t = 3 \)[/tex]:

[tex]\[ p = 600 \cdot \left(\frac{4}{5}\right)^3 = 307.2 \][/tex]

- At [tex]\( t = 4 \)[/tex]:

[tex]\[ p = 600 \cdot \left(\frac{4}{5}\right)^4 = 245.76 \][/tex]

Now, plotting these points:

[tex]\[ (0, 600), \quad (1, 480), \quad (2, 384), \quad (3, 307.2), \quad (4, 245.76) \][/tex]

Graph:

- On the x-axis, label the values from 0 to 4 (representing years since 2000).
- On the y-axis, label values from 0 to 600 (representing tree population in thousands).
- Plot the points and connect them to form the curve that shows how the population decays over time.

The graphical representation will show a decreasing trend over the years, illustrating how the tree population is shrinking according to the given model.