Question
Liz spent her summer break working at Wild Lakes waterpark. At the end of the summer, she put $570 of her earnings into a savings account that earns 1% interest each year.
Write an exponential equation in the form y=a(b)x that can model the amount in Liz's account, y, x years after starting the account.
Use whole numbers, decimals, or simplified fractions for the values of a and b.
y =
To the nearest dollar, how much will be in the savings account after 7 years if Liz makes no other deposits or withdrawals?
$
Write an exponential equation in the form y=a(b)x that can model the amount in Liz's account, y, x years after starting the account.
Use whole numbers, decimals, or simplified fractions for the values of a and b.
y =
To the nearest dollar, how much will be in the savings account after 7 years if Liz makes no other deposits or withdrawals?
$
Answers
GPT-4o mini
To model the amount in Liz's savings account using an exponential equation, we can use the formula:
\[ y = a(b)^x \]
Where:
- \( y \) is the amount in the account after \( x \) years.
- \( a \) is the initial amount (the principal).
- \( b \) is the growth factor.
- \( x \) is the number of years.
In this case:
- The initial amount \( a \) is $570.
- The interest rate is 1%, which as a decimal would be 0.01. The growth factor \( b \) would therefore be \( 1 + 0.01 = 1.01 \).
So, the equation modeling the amount in Liz's account becomes:
\[ y = 570(1.01)^x \]
Now, to find out how much will be in the account after 7 years, we can substitute \( x = 7 \) into the equation:
\[ y = 570(1.01)^7 \]
Calculating \( (1.01)^7 \):
\[ (1.01)^7 \approx 1.0714 \] (using a calculator)
Now substituting back into the equation:
\[ y \approx 570 \times 1.0714 \]
\[ y \approx 610.68 \]
To the nearest dollar, the amount in the savings account after 7 years will be:
\(\$611\)
So, the final answers are:
- The exponential equation: \( y = 570(1.01)^x \)
- The amount after 7 years: \$611.
\[ y = a(b)^x \]
Where:
- \( y \) is the amount in the account after \( x \) years.
- \( a \) is the initial amount (the principal).
- \( b \) is the growth factor.
- \( x \) is the number of years.
In this case:
- The initial amount \( a \) is $570.
- The interest rate is 1%, which as a decimal would be 0.01. The growth factor \( b \) would therefore be \( 1 + 0.01 = 1.01 \).
So, the equation modeling the amount in Liz's account becomes:
\[ y = 570(1.01)^x \]
Now, to find out how much will be in the account after 7 years, we can substitute \( x = 7 \) into the equation:
\[ y = 570(1.01)^7 \]
Calculating \( (1.01)^7 \):
\[ (1.01)^7 \approx 1.0714 \] (using a calculator)
Now substituting back into the equation:
\[ y \approx 570 \times 1.0714 \]
\[ y \approx 610.68 \]
To the nearest dollar, the amount in the savings account after 7 years will be:
\(\$611\)
So, the final answers are:
- The exponential equation: \( y = 570(1.01)^x \)
- The amount after 7 years: \$611.