To determine how long it will be before the balance in Lisa's checking and savings accounts are equal, we need to model the balances of both accounts over time. Let's break down the details given for both accounts and create our equations.
1. Checking Account:
- Initial deposit: \[tex]$50,000
- Average withdrawal rate: \$[/tex]2,500 per year
Since the balance decreases by \[tex]$2,500 each year, we can express the balance as:
\[
B_{\text{checking}} = 50,000 - 2,500t
\]
2. Savings Account:
- Initial deposit: \$[/tex]2,000
- Interest rate: continuously compounded at 5%
The balance in a continuously compounded interest account is given by the formula:
[tex]\[
B_{\text{savings}} = P e^{rt}
\][/tex]
where [tex]\( P \)[/tex] is the principal amount, [tex]\( r \)[/tex] is the interest rate, and [tex]\( t \)[/tex] is the time in years. For Lisa's savings account:
[tex]\[
B_{\text{savings}} = 2,000 e^{0.05t}
\][/tex]
Now we want to find out when the balances of both accounts will be equal. Thus, we set up the following system of equations:
[tex]\[
\begin{cases}
B = 50,000 - 2,500t \quad &\text{(Checking Account)} \\
B = 2,000 e^{0.05t} \quad &\text{(Savings Account)}
\end{cases}
\][/tex]
Given the choices provided, we need to select the system of equations that accurately represents this situation:
A. [tex]\(\left\{\begin{array}{l}B = 50,000 + 2,500t \\ B = 2,000 e^{0.03t}\end{array}\right.\)[/tex] - This is incorrect since the checking account balance should decrease, not increase, and the savings account has an incorrect interest rate.
B. [tex]\(\left\{\begin{array}{l}B = 50,000 - 2,500t \\ B = 2,000 e^{0.05t}\end{array}\right.\)[/tex] - This is correct. The checking account decreases correctly by \$2,500 each year, and the savings account earns interest at 5%.
C. [tex]\(\left\{\begin{array}{l}B = 50,000 - 2,000t \\ B = 2,500 e^{0.002t}\end{array}\right.\)[/tex] - This is incorrect since the checking account withdrawal rate is wrong and the savings account interest rate is incorrect.
D. [tex]\(\left\{\begin{array}{l}B = 50,000 - 2,500t \\ B = 2,000 + 100t\end{array}\right.\)[/tex] - This is incorrect since the savings account balance should grow exponentially, not linearly.
Therefore, the correct choice is:
B. [tex]\(\left\{\begin{array}{l}B = 50,000 - 2,500t \\ B = 2,000 e^{0.05t}\end{array}\right.\)[/tex]
