Rachel and Jeffery are both opening savings accounts. Rachel deposits $[tex]$1,500$[/tex] in a savings account that earns [tex]$1.5 \%$[/tex] interest, compounded annually. Jeffery deposits $[tex]$1,200$[/tex] in a savings account that earns [tex]$1 \%$[/tex] interest per year, compounded continuously.
If [tex]$y$[/tex] represents the account balance after [tex]$t$[/tex] years, which two equations form the system that best models this situation?
[tex]$y=1,500(1.015)^t$[/tex]
[tex]$y=1,500(1.5)^t$[/tex]
[tex]$y=1,500(0.985)^t$[/tex]
[tex]$y=1,200 e^t$[/tex]
[tex]$y=1,200 e^{0.01t}$[/tex]
[tex]$y=1,200 e^{1.01t}$[/tex]
Answer (2)
The equations that model Rachel's and Jeffery's savings accounts after t years are: Rachel's account is given by y = 1500 ( 1.015 ) t , and Jeffery's account by y = 1200 e 0.01 t .
;
Rachel's account balance is modeled by the equation y = 1500 ( 1.015 ) t .
Jeffery's account balance is modeled by the equation y = 1200 e 0.01 t .
These equations represent the growth of their savings accounts over time, considering compound interest.
The system of equations that best models this situation consists of y = 1500 ( 1.015 ) t and y = 1200 e 0.01 t .
Explanation
Problem Analysis Let's analyze the problem. Rachel's account earns interest compounded annually, while Jeffery's account earns interest compounded continuously. We need to find the equations that model their account balances after t years.
Rachel's Account For Rachel's account, the formula for annual compound interest is: y = P ( 1 + r ) t where:
y is the account balance after t years
P is the principal amount (initial deposit)
r is the annual interest rate (as a decimal)
t is the number of years
Rachel's Equation Rachel's principal is $1 , 500 and her interest rate is 1.5% = 0.015 . Plugging these values into the formula, we get: y = 1500 ( 1 + 0.015 ) t = 1500 ( 1.015 ) t
Jeffery's Account For Jeffery's account, the formula for continuous compound interest is: y = P e r t where:
y is the account balance after t years
P is the principal amount (initial deposit)
r is the annual interest rate (as a decimal)
t is the number of years
e is the base of the natural logarithm (approximately 2.71828)
Jeffery's Equation Jeffery's principal is $1 , 200 and his interest rate is 1% = 0.01 . Plugging these values into the formula, we get: y = 1200 e 0.01 t
Final Equations Therefore, the two equations that model the situation are: Rachel: y = 1500 ( 1.015 ) t Jeffery: y = 1200 e 0.01 t
Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, knowing how your investments grow over time with compound interest helps you estimate your future savings. Similarly, when taking out a loan, understanding the compound interest can help you assess the total cost of borrowing and compare different loan options. By mastering these concepts, you can make sound financial choices and achieve your long-term financial goals.
