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Sagot :
Let's analyze the table given for the population growth of Javan gibbons on three different islands over a period of six years. The three islands are Batanta, Misool, and Salawati.
We'll separate the data for each island and analyze the type of function that best represents each set of data.
1. Batanta Island (B(x)):
| Years (x) | Batanta [tex]\( B(x) \)[/tex] |
|-----------|--------------------|
| 0 | 2 |
| 1 | 6 |
| 2 | 18 |
| 3 | 54 |
| 4 | 162 |
| 5 | 486 |
To determine the type of function:
- From year 0 to year 1, the population goes from 2 to 6, which is a factor of 3.
- From year 1 to year 2, the population goes from 6 to 18, again a factor of 3.
- Continuing this pattern, each subsequent year the population seems to multiply by 3.
This multiplication suggests that the population [tex]\( B(x) \)[/tex] on Batanta Island follows a geometric progression, indicating an exponential function of the form [tex]\( B(x) = 2 \cdot 3^x \)[/tex].
2. Misool Island (M(x)):
| Years (x) | Misool [tex]\( M(x) \)[/tex] |
|-----------|--------------------|
| 0 | 20 |
| 1 | 120 |
| 2 | 420 |
| 3 | 920 |
| 4 | 1620 |
| 5 | 2520 |
An analysis of the differences (first differences):
- From year 0 to year 1: [tex]\( 120 - 20 = 100 \)[/tex]
- From year 1 to year 2: [tex]\( 420 - 120 = 300 \)[/tex]
- From year 2 to year 3: [tex]\( 920 - 420 = 500 \)[/tex]
- From year 3 to year 4: [tex]\( 1620 - 920 = 700 \)[/tex]
- From year 4 to year 5: [tex]\( 2520 - 1620 = 900 \)[/tex]
The first differences are not constant. Let's check the second differences:
- From 100 to 300: [tex]\( 300 - 100 = 200 \)[/tex]
- From 300 to 500: [tex]\( 500 - 300 = 200 \)[/tex]
- From 500 to 700: [tex]\( 700 - 500 = 200 \)[/tex]
- From 700 to 900: [tex]\( 900 - 700 = 200 \)[/tex]
The second differences are constant, indicating that [tex]\( M(x) \)[/tex] follows a quadratic function. Specifically, this can be modeled as [tex]\( M(x) = 100x^2 + 20 \)[/tex].
3. Salawati Island (S(x)):
| Years (x) | Salawati [tex]\( S(x) \)[/tex] |
|-----------|---------------------|
| 0 | 38 |
| 1 | 81 |
| 2 | 124 |
| 3 | 167 |
| 4 | 210 |
| 5 | 253 |
An analysis of the differences (first differences):
- From year 0 to year 1: [tex]\( 81 - 38 = 43 \)[/tex]
- From year 1 to year 2: [tex]\( 124 - 81 = 43 \)[/tex]
- From year 2 to year 3: [tex]\( 167 - 124 = 43 \)[/tex]
- From year 3 to year 4: [tex]\( 210 - 167 = 43 \)[/tex]
- From year 4 to year 5: [tex]\( 253 - 210 = 43 \)[/tex]
Since the first differences are consistent and constant, [tex]\( S(x) \)[/tex] follows a linear function. This function can be modeled as [tex]\( S(x) = 43x + 38 \)[/tex].
In conclusion:
- Batanta: [tex]\( B(x) = 2 \cdot 3^x \)[/tex] (Exponential Function)
- Misool: [tex]\( M(x) = 100x^2 + 20 \)[/tex] (Quadratic Function)
- Salawati: [tex]\( S(x) = 43x + 38 \)[/tex] (Linear Function)
These are the functional forms representing the population growth of Javan gibbons on the three islands over the specified time period.
We'll separate the data for each island and analyze the type of function that best represents each set of data.
1. Batanta Island (B(x)):
| Years (x) | Batanta [tex]\( B(x) \)[/tex] |
|-----------|--------------------|
| 0 | 2 |
| 1 | 6 |
| 2 | 18 |
| 3 | 54 |
| 4 | 162 |
| 5 | 486 |
To determine the type of function:
- From year 0 to year 1, the population goes from 2 to 6, which is a factor of 3.
- From year 1 to year 2, the population goes from 6 to 18, again a factor of 3.
- Continuing this pattern, each subsequent year the population seems to multiply by 3.
This multiplication suggests that the population [tex]\( B(x) \)[/tex] on Batanta Island follows a geometric progression, indicating an exponential function of the form [tex]\( B(x) = 2 \cdot 3^x \)[/tex].
2. Misool Island (M(x)):
| Years (x) | Misool [tex]\( M(x) \)[/tex] |
|-----------|--------------------|
| 0 | 20 |
| 1 | 120 |
| 2 | 420 |
| 3 | 920 |
| 4 | 1620 |
| 5 | 2520 |
An analysis of the differences (first differences):
- From year 0 to year 1: [tex]\( 120 - 20 = 100 \)[/tex]
- From year 1 to year 2: [tex]\( 420 - 120 = 300 \)[/tex]
- From year 2 to year 3: [tex]\( 920 - 420 = 500 \)[/tex]
- From year 3 to year 4: [tex]\( 1620 - 920 = 700 \)[/tex]
- From year 4 to year 5: [tex]\( 2520 - 1620 = 900 \)[/tex]
The first differences are not constant. Let's check the second differences:
- From 100 to 300: [tex]\( 300 - 100 = 200 \)[/tex]
- From 300 to 500: [tex]\( 500 - 300 = 200 \)[/tex]
- From 500 to 700: [tex]\( 700 - 500 = 200 \)[/tex]
- From 700 to 900: [tex]\( 900 - 700 = 200 \)[/tex]
The second differences are constant, indicating that [tex]\( M(x) \)[/tex] follows a quadratic function. Specifically, this can be modeled as [tex]\( M(x) = 100x^2 + 20 \)[/tex].
3. Salawati Island (S(x)):
| Years (x) | Salawati [tex]\( S(x) \)[/tex] |
|-----------|---------------------|
| 0 | 38 |
| 1 | 81 |
| 2 | 124 |
| 3 | 167 |
| 4 | 210 |
| 5 | 253 |
An analysis of the differences (first differences):
- From year 0 to year 1: [tex]\( 81 - 38 = 43 \)[/tex]
- From year 1 to year 2: [tex]\( 124 - 81 = 43 \)[/tex]
- From year 2 to year 3: [tex]\( 167 - 124 = 43 \)[/tex]
- From year 3 to year 4: [tex]\( 210 - 167 = 43 \)[/tex]
- From year 4 to year 5: [tex]\( 253 - 210 = 43 \)[/tex]
Since the first differences are consistent and constant, [tex]\( S(x) \)[/tex] follows a linear function. This function can be modeled as [tex]\( S(x) = 43x + 38 \)[/tex].
In conclusion:
- Batanta: [tex]\( B(x) = 2 \cdot 3^x \)[/tex] (Exponential Function)
- Misool: [tex]\( M(x) = 100x^2 + 20 \)[/tex] (Quadratic Function)
- Salawati: [tex]\( S(x) = 43x + 38 \)[/tex] (Linear Function)
These are the functional forms representing the population growth of Javan gibbons on the three islands over the specified time period.
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