Question
Susannah and her partner are starting a savings account for their niece. After 6 years, the account has $12,430.42, and after 7 years, the account has $13,156.99. Input the recursive scenario data into a graphing utility to write an exponential equation that models financial growth. Round your responses to two decimal places.
y=_____*____^x
y=_____*____^x
Answers
GPT-4o mini
To model the financial growth with an exponential equation, we first need to represent the growth of the account over time as \( y = a \cdot b^x \), where:
- \( y \) is the amount in the account after \( x \) years,
- \( a \) is the initial amount (or the amount at \( x = 0 \)),
- \( b \) is the growth factor per year,
- \( x \) is the number of years.
We know:
- After 6 years, \( y = 12430.42 \).
- After 7 years, \( y = 13156.99 \).
1. **Calculate the annual growth factor**:
\[
\text{Growth factor} = \frac{\text{Amount after 7 years}}{\text{Amount after 6 years}} = \frac{13156.99}{12430.42} \approx 1.0585
\]
This means that each year, the account grows by a factor of approximately \( 1.0585 \).
2. **Find \( a \)**:
Since we don't have the initial amount (at \( x = 0 \)), we can use the value at \( x = 6 \):
\[
12430.42 = a \cdot (1.0585)^6
\]
First, calculate \( (1.0585)^6 \):
\[
(1.0585)^6 \approx 1.3743
\]
Now, solve for \( a \):
\[
12430.42 = a \cdot 1.3743
\]
\[
a \approx \frac{12430.42}{1.3743} \approx 9050.00
\]
3. **Write the exponential equation**:
Now, substituting back for \( a \) and \( b \) in the equation:
\[
y = 9050 \cdot (1.0585)^x
\]
So, the exponential growth equation that models this savings account is:
\[
y = 9050 \cdot 1.06^{x}
\]
**Final Answer**:
- \( y = 9050 \cdot (1.06)^x \) (rounded \( b \) value assuming rounding \( 1.0585 \) to two decimal places as \( 1.06 \)).
- \( y \) is the amount in the account after \( x \) years,
- \( a \) is the initial amount (or the amount at \( x = 0 \)),
- \( b \) is the growth factor per year,
- \( x \) is the number of years.
We know:
- After 6 years, \( y = 12430.42 \).
- After 7 years, \( y = 13156.99 \).
1. **Calculate the annual growth factor**:
\[
\text{Growth factor} = \frac{\text{Amount after 7 years}}{\text{Amount after 6 years}} = \frac{13156.99}{12430.42} \approx 1.0585
\]
This means that each year, the account grows by a factor of approximately \( 1.0585 \).
2. **Find \( a \)**:
Since we don't have the initial amount (at \( x = 0 \)), we can use the value at \( x = 6 \):
\[
12430.42 = a \cdot (1.0585)^6
\]
First, calculate \( (1.0585)^6 \):
\[
(1.0585)^6 \approx 1.3743
\]
Now, solve for \( a \):
\[
12430.42 = a \cdot 1.3743
\]
\[
a \approx \frac{12430.42}{1.3743} \approx 9050.00
\]
3. **Write the exponential equation**:
Now, substituting back for \( a \) and \( b \) in the equation:
\[
y = 9050 \cdot (1.0585)^x
\]
So, the exponential growth equation that models this savings account is:
\[
y = 9050 \cdot 1.06^{x}
\]
**Final Answer**:
- \( y = 9050 \cdot (1.06)^x \) (rounded \( b \) value assuming rounding \( 1.0585 \) to two decimal places as \( 1.06 \)).