Answer :
Let's delve into the given function and analyze the statements step by step.
Given:
[tex]\[ P(t) = 3 \ln (2t + 3) - 5 \][/tex]
### 1. Initial Height of the Peach Tree
To find the initial height of the peach tree, we substitute [tex]\(t = 0\)[/tex] into the given function:
[tex]\[ P(0) = 3 \ln (2(0) + 3) - 5 = 3 \ln 3 - 5 \][/tex]
Since [tex]\(\ln 3\)[/tex] is a positive number, we know that [tex]\( 3 \ln 3 > 0\)[/tex]. Therefore, [tex]\( 3 \ln 3 - 5 \)[/tex] is a positive number subtracted by 5, which may be positive or negative.
Thus, the initial height is [tex]\( 3 \ln 3 - 5 \)[/tex].
So, the first statement, "The avocado seed was planted at ground level," would need us to have more information about the avocado tree and cannot be determined solely from the given function for the peach tree. Usually, if the height function closely examines the start (including precise values known for both), we can conclude accurately.
### 2. Peach Tree Eventually Being Taller
To determine if the peach tree will be taller than the avocado tree, we would need a function for the avocado tree to compare. Since we only have the peach tree's function, we cannot accurately decide on this statement either.
### 3. Peach Seed was Planted Closer to Ground Level
This question is ambiguous without knowing the height at the initial planting time. If the height starts low enough (e.g., near zero) and given that initial observation [tex]\[3\ln 3\][/tex] is approximately [tex]\( \approx 3.295\)[/tex] decimal value, so [tex]\([3*3.295 - 5]=4.885 - 5= -0.105). This deduction helps us deduce zero or negative(some initial seed planting depth) but we need exact comparison basis. Full statement about avocado tree detail on planting depth to make clear decision. ### 4. Average Rate of Change for Avocado Tree The average rate of change for the avocado tree between the 2nd and 6th weeks cannot be determined as we don't have the function or data for the avocado tree. ### 5. Peach Tree's Average Rate of Change We calculate the heights of the peach tree at \(t = 2\)[/tex] and [tex]\(t = 6\)[/tex]:
For [tex]\( t = 2 \)[/tex]:
[tex]\[ P(2) = 3 \ln (2(2) + 3) - 5 = 3 \ln 7 - 5 \][/tex]
For [tex]\( t = 6 \)[/tex]:
[tex]\[ P(6) = 3 \ln (2(6) + 3) - 5 = 3 \ln 15 - 5 \][/tex]
The average rate of change between the 2nd and 6th weeks can be calculated as:
[tex]\[ \text{Average Rate of Change} = \frac{P(6) - P(2)}{6 - 2} = \frac{(3 \ln 15 - 5) - (3 \ln 7 - 5)}{4} = \frac{3 \ln 15 - 3 \ln 7}{4} = \frac{3 (\ln 15 - \ln 7)}{4} = \frac{3 \ln \frac{15}{7}}{4} \approx \frac{3 \cdot 1.317}{4} = \frac{3.951}{4} \approx 0.988 \][/tex]
### Conclusion about the Statements:
1. Cannot definitively say if avocado seed was at ground level without comparison.
2. Cannot determine if the peach tree will eventually be taller than the avocado tree without more data.
3. Based on derived height [tex]\([3Ln3 - 5] \)[/tex], if less than zero then supposed closer to ground(initial).
4. Cannot determine average rate of change for the avocado tree without more data.
5. Average rate of change of the peach tree between the 2nd and 6th weeks is approximately 0.988.
So considering all aspects, the 5th statement 'The peach tree had a greater average rate of change between the 2nd and 6th weeks after being planted' is likely correct for peach tree context specifically assuming comparison applies similarly.
Given:
[tex]\[ P(t) = 3 \ln (2t + 3) - 5 \][/tex]
### 1. Initial Height of the Peach Tree
To find the initial height of the peach tree, we substitute [tex]\(t = 0\)[/tex] into the given function:
[tex]\[ P(0) = 3 \ln (2(0) + 3) - 5 = 3 \ln 3 - 5 \][/tex]
Since [tex]\(\ln 3\)[/tex] is a positive number, we know that [tex]\( 3 \ln 3 > 0\)[/tex]. Therefore, [tex]\( 3 \ln 3 - 5 \)[/tex] is a positive number subtracted by 5, which may be positive or negative.
Thus, the initial height is [tex]\( 3 \ln 3 - 5 \)[/tex].
So, the first statement, "The avocado seed was planted at ground level," would need us to have more information about the avocado tree and cannot be determined solely from the given function for the peach tree. Usually, if the height function closely examines the start (including precise values known for both), we can conclude accurately.
### 2. Peach Tree Eventually Being Taller
To determine if the peach tree will be taller than the avocado tree, we would need a function for the avocado tree to compare. Since we only have the peach tree's function, we cannot accurately decide on this statement either.
### 3. Peach Seed was Planted Closer to Ground Level
This question is ambiguous without knowing the height at the initial planting time. If the height starts low enough (e.g., near zero) and given that initial observation [tex]\[3\ln 3\][/tex] is approximately [tex]\( \approx 3.295\)[/tex] decimal value, so [tex]\([3*3.295 - 5]=4.885 - 5= -0.105). This deduction helps us deduce zero or negative(some initial seed planting depth) but we need exact comparison basis. Full statement about avocado tree detail on planting depth to make clear decision. ### 4. Average Rate of Change for Avocado Tree The average rate of change for the avocado tree between the 2nd and 6th weeks cannot be determined as we don't have the function or data for the avocado tree. ### 5. Peach Tree's Average Rate of Change We calculate the heights of the peach tree at \(t = 2\)[/tex] and [tex]\(t = 6\)[/tex]:
For [tex]\( t = 2 \)[/tex]:
[tex]\[ P(2) = 3 \ln (2(2) + 3) - 5 = 3 \ln 7 - 5 \][/tex]
For [tex]\( t = 6 \)[/tex]:
[tex]\[ P(6) = 3 \ln (2(6) + 3) - 5 = 3 \ln 15 - 5 \][/tex]
The average rate of change between the 2nd and 6th weeks can be calculated as:
[tex]\[ \text{Average Rate of Change} = \frac{P(6) - P(2)}{6 - 2} = \frac{(3 \ln 15 - 5) - (3 \ln 7 - 5)}{4} = \frac{3 \ln 15 - 3 \ln 7}{4} = \frac{3 (\ln 15 - \ln 7)}{4} = \frac{3 \ln \frac{15}{7}}{4} \approx \frac{3 \cdot 1.317}{4} = \frac{3.951}{4} \approx 0.988 \][/tex]
### Conclusion about the Statements:
1. Cannot definitively say if avocado seed was at ground level without comparison.
2. Cannot determine if the peach tree will eventually be taller than the avocado tree without more data.
3. Based on derived height [tex]\([3Ln3 - 5] \)[/tex], if less than zero then supposed closer to ground(initial).
4. Cannot determine average rate of change for the avocado tree without more data.
5. Average rate of change of the peach tree between the 2nd and 6th weeks is approximately 0.988.
So considering all aspects, the 5th statement 'The peach tree had a greater average rate of change between the 2nd and 6th weeks after being planted' is likely correct for peach tree context specifically assuming comparison applies similarly.
