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Sagot :
Let's analyze the given information step by step.
We have the recorded rainfall data over time:
[tex]\[ \begin{array}{|c|c|} \hline \text{Time (hr)} & \text{Rainfall (in.)} \\ \hline 1 & 1.65 \\ \hline 2 & 3.30 \\ \hline 3 & 4.95 \\ \hline 4 & 6.60 \\ \hline \end{array} \][/tex]
We are given two functions to test:
1. Anna proposes [tex]\( R(h) = 1.65^h \)[/tex]
2. Luke proposes [tex]\( R(h) = 1.65h \)[/tex]
Anna's Function: [tex]\( R(h) = 1.65^h \)[/tex]
We will calculate the rainfall according to Anna's function for each hour:
- For [tex]\( h = 1 \)[/tex], [tex]\( R(1) = 1.65^1 = 1.65 \)[/tex]
- For [tex]\( h = 2 \)[/tex], [tex]\( R(2) = 1.65^2 \approx 2.7225 \)[/tex]
- For [tex]\( h = 3 \)[/tex], [tex]\( R(3) = 1.65^3 \approx 4.4921 \)[/tex]
- For [tex]\( h = 4 \)[/tex], [tex]\( R(4) = 1.65^4 \approx 7.4120 \)[/tex]
When we compare these values with the actual recorded data, we notice that Anna's function does not match:
[tex]\[ \begin{array}{|c|c|} \hline \text{Time (hr)} & \text{Rainfall (in.)} & \text{Anna's Function} \\ \hline 1 & 1.65 & 1.65 \\ \hline 2 & 3.30 & 2.7225 \\ \hline 3 & 4.95 & 4.4921 \\ \hline 4 & 6.60 & 7.4120 \\ \hline \end{array} \][/tex]
Therefore, Anna's function [tex]\(\boxed{\text{cannot}}\)[/tex] be used to find the amount of rainfall after [tex]\( h \)[/tex] hours. The function displays exponential growth, not a consistent linear increase. Thus, each hour, an additional [tex]\(\boxed{\text{N/A}}\)[/tex] inches of rain is accumulated. Equal differences represent a [tex]\(\boxed{\text{N/A}}\)[/tex] relationship.
Luke's Function: [tex]\( R(h) = 1.65h \)[/tex]
Next, we calculate the rainfall according to Luke's function for each hour:
- For [tex]\( h = 1 \)[/tex], [tex]\( R(1) = 1.65 \times 1 = 1.65 \)[/tex]
- For [tex]\( h = 2 \)[/tex], [tex]\( R(2) = 1.65 \times 2 = 3.30 \)[/tex]
- For [tex]\( h = 3 \)[/tex], [tex]\( R(3) = 1.65 \times 3 = 4.95 \)[/tex]
- For [tex]\( h = 4 \)[/tex], [tex]\( R(4) = 1.65 \times 4 = 6.60 \)[/tex]
When we compare these values with the actual recorded data, we notice that Luke's function matches perfectly:
[tex]\[ \begin{array}{|c|c|} \hline \text{Time (hr)} & \text{Rainfall (in.)} & \text{Luke's Function} \\ \hline 1 & 1.65 & 1.65 \\ \hline 2 & 3.30 & 3.30 \\ \hline 3 & 4.95 & 4.95 \\ \hline 4 & 6.60 & 6.60 \\ \hline \end{array} \][/tex]
Therefore, Luke's function [tex]\(\boxed{\text{can}}\)[/tex] be used to find the amount of rainfall after [tex]\( h \)[/tex] hours. For every hour that passes, [tex]\(\boxed{1.65}\)[/tex] inches of rain is accumulated. This represents a consistent, linear relationship.
In summary:
Anna [tex]\( R(h) = 1.65^h \)[/tex]
- This function [tex]\(\boxed{\text{cannot}}\)[/tex] be used to find the amount of rainfall after [tex]\( h \)[/tex] hours.
- Each hour, an additional [tex]\(\boxed{\text{N/A}}\)[/tex] inches of rain is accumulated.
- Equal differences represent a [tex]\(\boxed{\text{N/A}}\)[/tex] relationship.
Luke [tex]\( R(h) = 1.65h \)[/tex]
- This function [tex]\(\boxed{\text{can}}\)[/tex] be used to find the amount of rainfall after [tex]\( h \)[/tex] hours.
- For every hour that passes, [tex]\(\boxed{1.65}\)[/tex] inches of rain is accumulated.
We have the recorded rainfall data over time:
[tex]\[ \begin{array}{|c|c|} \hline \text{Time (hr)} & \text{Rainfall (in.)} \\ \hline 1 & 1.65 \\ \hline 2 & 3.30 \\ \hline 3 & 4.95 \\ \hline 4 & 6.60 \\ \hline \end{array} \][/tex]
We are given two functions to test:
1. Anna proposes [tex]\( R(h) = 1.65^h \)[/tex]
2. Luke proposes [tex]\( R(h) = 1.65h \)[/tex]
Anna's Function: [tex]\( R(h) = 1.65^h \)[/tex]
We will calculate the rainfall according to Anna's function for each hour:
- For [tex]\( h = 1 \)[/tex], [tex]\( R(1) = 1.65^1 = 1.65 \)[/tex]
- For [tex]\( h = 2 \)[/tex], [tex]\( R(2) = 1.65^2 \approx 2.7225 \)[/tex]
- For [tex]\( h = 3 \)[/tex], [tex]\( R(3) = 1.65^3 \approx 4.4921 \)[/tex]
- For [tex]\( h = 4 \)[/tex], [tex]\( R(4) = 1.65^4 \approx 7.4120 \)[/tex]
When we compare these values with the actual recorded data, we notice that Anna's function does not match:
[tex]\[ \begin{array}{|c|c|} \hline \text{Time (hr)} & \text{Rainfall (in.)} & \text{Anna's Function} \\ \hline 1 & 1.65 & 1.65 \\ \hline 2 & 3.30 & 2.7225 \\ \hline 3 & 4.95 & 4.4921 \\ \hline 4 & 6.60 & 7.4120 \\ \hline \end{array} \][/tex]
Therefore, Anna's function [tex]\(\boxed{\text{cannot}}\)[/tex] be used to find the amount of rainfall after [tex]\( h \)[/tex] hours. The function displays exponential growth, not a consistent linear increase. Thus, each hour, an additional [tex]\(\boxed{\text{N/A}}\)[/tex] inches of rain is accumulated. Equal differences represent a [tex]\(\boxed{\text{N/A}}\)[/tex] relationship.
Luke's Function: [tex]\( R(h) = 1.65h \)[/tex]
Next, we calculate the rainfall according to Luke's function for each hour:
- For [tex]\( h = 1 \)[/tex], [tex]\( R(1) = 1.65 \times 1 = 1.65 \)[/tex]
- For [tex]\( h = 2 \)[/tex], [tex]\( R(2) = 1.65 \times 2 = 3.30 \)[/tex]
- For [tex]\( h = 3 \)[/tex], [tex]\( R(3) = 1.65 \times 3 = 4.95 \)[/tex]
- For [tex]\( h = 4 \)[/tex], [tex]\( R(4) = 1.65 \times 4 = 6.60 \)[/tex]
When we compare these values with the actual recorded data, we notice that Luke's function matches perfectly:
[tex]\[ \begin{array}{|c|c|} \hline \text{Time (hr)} & \text{Rainfall (in.)} & \text{Luke's Function} \\ \hline 1 & 1.65 & 1.65 \\ \hline 2 & 3.30 & 3.30 \\ \hline 3 & 4.95 & 4.95 \\ \hline 4 & 6.60 & 6.60 \\ \hline \end{array} \][/tex]
Therefore, Luke's function [tex]\(\boxed{\text{can}}\)[/tex] be used to find the amount of rainfall after [tex]\( h \)[/tex] hours. For every hour that passes, [tex]\(\boxed{1.65}\)[/tex] inches of rain is accumulated. This represents a consistent, linear relationship.
In summary:
Anna [tex]\( R(h) = 1.65^h \)[/tex]
- This function [tex]\(\boxed{\text{cannot}}\)[/tex] be used to find the amount of rainfall after [tex]\( h \)[/tex] hours.
- Each hour, an additional [tex]\(\boxed{\text{N/A}}\)[/tex] inches of rain is accumulated.
- Equal differences represent a [tex]\(\boxed{\text{N/A}}\)[/tex] relationship.
Luke [tex]\( R(h) = 1.65h \)[/tex]
- This function [tex]\(\boxed{\text{can}}\)[/tex] be used to find the amount of rainfall after [tex]\( h \)[/tex] hours.
- For every hour that passes, [tex]\(\boxed{1.65}\)[/tex] inches of rain is accumulated.
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