Question
Jayden opens a savings account and deposits $100. If his annual interest rate is 4%, write an exponential function to model the situation. What is the exponential growth/decay factor? (b value)
Answers
GPT-4o mini
To model the situation of Jayden's savings account, we can use the formula for exponential growth:
\[ A(t) = A_0(1 + r)^t \]
Where:
- \( A(t) \) is the amount of money in the account after time \( t \),
- \( A_0 \) is the initial amount of money (the initial deposit),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years.
In this case:
- \( A_0 = 100 \) (the initial deposit),
- \( r = 0.04 \) (4% as a decimal).
Substituting these values into the formula, we get:
\[ A(t) = 100(1 + 0.04)^t \]
\[ A(t) = 100(1.04)^t \]
The exponential function that models the situation is:
\[ A(t) = 100(1.04)^t \]
Now, the exponential growth factor (the \( b \) value) in this function is:
**\( b = 1.04 \)**
This means that for each year that passes, the amount in the account increases by a factor of 1.04.
\[ A(t) = A_0(1 + r)^t \]
Where:
- \( A(t) \) is the amount of money in the account after time \( t \),
- \( A_0 \) is the initial amount of money (the initial deposit),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years.
In this case:
- \( A_0 = 100 \) (the initial deposit),
- \( r = 0.04 \) (4% as a decimal).
Substituting these values into the formula, we get:
\[ A(t) = 100(1 + 0.04)^t \]
\[ A(t) = 100(1.04)^t \]
The exponential function that models the situation is:
\[ A(t) = 100(1.04)^t \]
Now, the exponential growth factor (the \( b \) value) in this function is:
**\( b = 1.04 \)**
This means that for each year that passes, the amount in the account increases by a factor of 1.04.