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Answer :
To solve the problem, we need to find the function that represents the value of Gwen's savings account after [tex]\( x \)[/tex] years, given a principal amount of [tex]$7000 and an annual interest rate of 2.5%. Then, we will calculate her balance after 9 years.
1. Identifying the Correct Function:
With compound interest, the formula to calculate the future value \( f(x) \) of an investment is:
\[
f(x) = P \cdot (1 + r)^x
\]
where \( P \) is the principal amount, \( r \) is the annual interest rate (in decimal), and \( x \) is the number of years.
Given:
- Principal, \( P = 7000 \)
- Annual interest rate, \( r = 2.5\% = 0.025 \)
The correct function will be:
\[
f(x) = 7000 \cdot (1 + 0.025)^x = 7000 \cdot (1.025)^x
\]
From the options provided, this corresponds to:
(B) \( f(x) = 7000 \cdot (1.025)^x \)
2. Calculating the Balance After 9 Years:
We substitute \( x = 9 \) into the function to find the balance after 9 years:
\[
f(9) = 7000 \cdot (1.025)^9
\]
After performing the calculation, the balance after 9 years is approximately \$[/tex]8742.04.
So, Gwen's balance after 9 years will be approximately \$8742.04.
1. Identifying the Correct Function:
With compound interest, the formula to calculate the future value \( f(x) \) of an investment is:
\[
f(x) = P \cdot (1 + r)^x
\]
where \( P \) is the principal amount, \( r \) is the annual interest rate (in decimal), and \( x \) is the number of years.
Given:
- Principal, \( P = 7000 \)
- Annual interest rate, \( r = 2.5\% = 0.025 \)
The correct function will be:
\[
f(x) = 7000 \cdot (1 + 0.025)^x = 7000 \cdot (1.025)^x
\]
From the options provided, this corresponds to:
(B) \( f(x) = 7000 \cdot (1.025)^x \)
2. Calculating the Balance After 9 Years:
We substitute \( x = 9 \) into the function to find the balance after 9 years:
\[
f(9) = 7000 \cdot (1.025)^9
\]
After performing the calculation, the balance after 9 years is approximately \$[/tex]8742.04.
So, Gwen's balance after 9 years will be approximately \$8742.04.
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