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Sagot :
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.
### 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.
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