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Answer :
To determine the value of Gwen's savings account after a certain number of years with interest, we need to identify the correct function that represents the balance over time. The problem provides options for us to choose from:
1. Understanding the Function:
- Gwen deposits [tex]$7000 in a savings account with an annual interest rate of 2.5%.
- The interest is compounded annually, which means each year the interest is calculated on the new balance.
2. Exponential Growth Function:
- The function that represents the balance after \( x \) years with an interest rate of 2.5% is:
\[
f(x) = \text{initial amount} \times (1 + \text{interest rate})^x
\]
- Here, the initial amount is $[/tex]7000, and the interest rate is 2.5%, which is 0.025 as a decimal. Therefore, the function becomes:
[tex]\[
f(x) = 7000 \times (1 + 0.025)^x = 7000 \times (1.025)^x
\][/tex]
3. Identifying the Correct Option:
- We look for the option that matches this expression. Option (B) is correct:
[tex]\[
f(x) = 7000 \cdot (1.025)^x
\][/tex]
4. Calculating the Balance After 9 Years:
- Using the formula [tex]\( f(x) = 7000 \times (1.025)^x \)[/tex], we replace [tex]\( x \)[/tex] with 9 to find the balance after 9 years:
[tex]\[
f(9) = 7000 \times (1.025)^9
\][/tex]
5. Final Balance:
- After performing the calculation for 9 years, Gwen's account balance is approximately \[tex]$8742.04.
Therefore, her balance after 9 years will be approximately \$[/tex]8742.04.
1. Understanding the Function:
- Gwen deposits [tex]$7000 in a savings account with an annual interest rate of 2.5%.
- The interest is compounded annually, which means each year the interest is calculated on the new balance.
2. Exponential Growth Function:
- The function that represents the balance after \( x \) years with an interest rate of 2.5% is:
\[
f(x) = \text{initial amount} \times (1 + \text{interest rate})^x
\]
- Here, the initial amount is $[/tex]7000, and the interest rate is 2.5%, which is 0.025 as a decimal. Therefore, the function becomes:
[tex]\[
f(x) = 7000 \times (1 + 0.025)^x = 7000 \times (1.025)^x
\][/tex]
3. Identifying the Correct Option:
- We look for the option that matches this expression. Option (B) is correct:
[tex]\[
f(x) = 7000 \cdot (1.025)^x
\][/tex]
4. Calculating the Balance After 9 Years:
- Using the formula [tex]\( f(x) = 7000 \times (1.025)^x \)[/tex], we replace [tex]\( x \)[/tex] with 9 to find the balance after 9 years:
[tex]\[
f(9) = 7000 \times (1.025)^9
\][/tex]
5. Final Balance:
- After performing the calculation for 9 years, Gwen's account balance is approximately \[tex]$8742.04.
Therefore, her balance after 9 years will be approximately \$[/tex]8742.04.
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