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Sagot :
### Part A
To determine whether the amount of money in account [tex]\( A \)[/tex] is increasing or decreasing, we analyze the function given:
[tex]\[ f(x) = 9.628(0.92)^x \][/tex]
Here, the base of the exponential function is [tex]\( 0.92 \)[/tex]. Since [tex]\( 0.92 \)[/tex] is less than 1, this indicates that the function is decreasing. In this context, each year the amount of money in account [tex]\( A \)[/tex] decreases.
To find the percentage decrease per year, we observe the base of the exponent:
[tex]\[ \text{Base} = 0.92 \][/tex]
The percentage decrease per year can be calculated as follows:
[tex]\[ \text{Percentage Decrease per Year} = (1 - \text{Base}) \times 100 \][/tex]
[tex]\[ \text{Percentage Decrease per Year} = (1 - 0.92) \times 100 \][/tex]
[tex]\[ \text{Percentage Decrease per Year} = 0.08 \times 100 \][/tex]
[tex]\[ \text{Percentage Decrease per Year} = 8\% \][/tex]
Thus, the amount of money in account [tex]\( A \)[/tex] is decreasing by 8% per year.
### Part B
The table shows the amount of money in account [tex]\( B \)[/tex] over different years:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline r \text{ (number of years)} & 1 & 2 & 3 & 4 \\ \hline g(r) \text{ (amount in dollars)} & 8,972 & 8,074.80 & 7,267.32 & 6,540.59 \\ \hline \end{array} \][/tex]
We will calculate the percentage change for each year in account [tex]\( B \)[/tex]:
From year 1 to year 2:
[tex]\[ \text{Percentage Change} = \left( \frac{8972 - 8074.80}{8972} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = \left( \frac{8972 - 8074.80}{8972} \right) \times 100 \approx 10\% \][/tex]
From year 2 to year 3:
[tex]\[ \text{Percentage Change} = \left( \frac{8074.80 - 7267.32}{8074.80} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = \left( \frac{8074.80 - 7267.32}{8074.80} \right) \times 100 \approx 10\% \][/tex]
From year 3 to year 4:
[tex]\[ \text{Percentage Change} = \left( \frac{7267.32 - 6540.59}{7267.32} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = \left( \frac{7267.32 - 6540.59}{7267.32} \right) \times 100 \approx 10\% \][/tex]
Here are the percentage changes for each year:
- Year 1 to 2: 10%
- Year 2 to 3: 10%
- Year 3 to 4: 10%
The greatest percentage change in the amount of money in account [tex]\( B \)[/tex] over the previous years is [tex]\( 10\% \)[/tex].
### Conclusion:
- In Part A, the amount of money in account [tex]\( A \)[/tex] is decreasing by 8% per year.
- In Part B, the account [tex]\( B \)[/tex] recorded a greater percentage change of 10% in the amount of money over the previous year.
To determine whether the amount of money in account [tex]\( A \)[/tex] is increasing or decreasing, we analyze the function given:
[tex]\[ f(x) = 9.628(0.92)^x \][/tex]
Here, the base of the exponential function is [tex]\( 0.92 \)[/tex]. Since [tex]\( 0.92 \)[/tex] is less than 1, this indicates that the function is decreasing. In this context, each year the amount of money in account [tex]\( A \)[/tex] decreases.
To find the percentage decrease per year, we observe the base of the exponent:
[tex]\[ \text{Base} = 0.92 \][/tex]
The percentage decrease per year can be calculated as follows:
[tex]\[ \text{Percentage Decrease per Year} = (1 - \text{Base}) \times 100 \][/tex]
[tex]\[ \text{Percentage Decrease per Year} = (1 - 0.92) \times 100 \][/tex]
[tex]\[ \text{Percentage Decrease per Year} = 0.08 \times 100 \][/tex]
[tex]\[ \text{Percentage Decrease per Year} = 8\% \][/tex]
Thus, the amount of money in account [tex]\( A \)[/tex] is decreasing by 8% per year.
### Part B
The table shows the amount of money in account [tex]\( B \)[/tex] over different years:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline r \text{ (number of years)} & 1 & 2 & 3 & 4 \\ \hline g(r) \text{ (amount in dollars)} & 8,972 & 8,074.80 & 7,267.32 & 6,540.59 \\ \hline \end{array} \][/tex]
We will calculate the percentage change for each year in account [tex]\( B \)[/tex]:
From year 1 to year 2:
[tex]\[ \text{Percentage Change} = \left( \frac{8972 - 8074.80}{8972} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = \left( \frac{8972 - 8074.80}{8972} \right) \times 100 \approx 10\% \][/tex]
From year 2 to year 3:
[tex]\[ \text{Percentage Change} = \left( \frac{8074.80 - 7267.32}{8074.80} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = \left( \frac{8074.80 - 7267.32}{8074.80} \right) \times 100 \approx 10\% \][/tex]
From year 3 to year 4:
[tex]\[ \text{Percentage Change} = \left( \frac{7267.32 - 6540.59}{7267.32} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = \left( \frac{7267.32 - 6540.59}{7267.32} \right) \times 100 \approx 10\% \][/tex]
Here are the percentage changes for each year:
- Year 1 to 2: 10%
- Year 2 to 3: 10%
- Year 3 to 4: 10%
The greatest percentage change in the amount of money in account [tex]\( B \)[/tex] over the previous years is [tex]\( 10\% \)[/tex].
### Conclusion:
- In Part A, the amount of money in account [tex]\( A \)[/tex] is decreasing by 8% per year.
- In Part B, the account [tex]\( B \)[/tex] recorded a greater percentage change of 10% in the amount of money over the previous year.
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