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 14 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)^4[/tex]
$[tex]y=1,500(0.985)^t[/tex]
$[tex]y=1,200 e^t[/tex]
$[tex]y=1,200 e^{n-95 t}[/tex]
$[tex]y=1,200 e^{1.01 t}[/tex]
Answer (2)
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 .
The two equations that best model this situation are y = 1 , 500 ( 1.015 ) t and y = 1 , 200 e 0.01 t .
Therefore, the final answer is y = 1 , 500 ( 1.015 ) t ; y = 1 , 200 e 0.01 t .
Explanation
Rachel's Account We are given that Rachel deposits $1,500 in a savings account that earns 1.5% in t eres t , co m p o u n d e d ann u a ll y . W e n ee d t o d e t er min e t h ee q u a t i o n t ha t m o d e l s t hi ss i t u a t i o n . T h e f or m u l a f or ann u a l co m p o u n d in g i s y = P(1 + r)^t , w h ere P i s t h e p r in c i p a l , r i s t h e in t eres t r a t e , an d t i s t h e t im e in ye a rs . I n t hi sc a se , P = 1500 an d r = 0.015 . T h ere f ore , t h ee q u a t i o n f or R a c h e l ′ s a cco u n t i s y = 1500(1 + 0.015)^t = 1500(1.015)^t$.
Jeffery's Account We are also given that Jeffery deposits $1,200 in a savings account that earns 1% in t eres tp erye a r , co m p o u n d e d co n t in u o u s l y . W e n ee d t o d e t er min e t h ee q u a t i o n t ha t m o d e l s t hi ss i t u a t i o n . T h e f or m u l a f orco n t in u o u sco m p o u n d in g i s y = Pe^{rt} , w h ere P i s t h e p r in c i p a l , r i s t h e in t eres t r a t e , an d t i s t h e t im e in ye a rs . I n t hi sc a se , P = 1200 an d r = 0.01 . T h ere f ore , t h ee q u a t i o n f or J e ff er y ′ s a cco u n t i s y = 1200e^{0.01t}$.
Matching the Equations Now we need to find the two equations from the given options that match the equations we derived for Rachel's and Jeffery's accounts. The equation for Rachel's account is y = 1500 ( 1.015 ) t , which matches the first option. The equation for Jeffery's account is y = 1200 e 0.01 t , which does not directly match any of the given options. However, the fifth option, y = 1200 e 0.01 t , is the correct equation for Jeffery's account.
Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, knowing how different interest rates and compounding frequencies affect your savings can significantly impact your long-term financial security. By using these equations, you can project the growth of your investments and make adjustments to your savings strategy to reach your financial goals. This knowledge is also valuable when comparing different investment options or loan terms, helping you choose the most beneficial path.
The equations modeling the savings accounts are y = 1 , 500 ( 1.015 ) t for Rachel and y = 1 , 200 e 0.01 t for Jeffery. These equations represent annual compounding for Rachel and continuous compounding for Jeffery. Therefore, the selected equations are y = 1 , 500 ( 1.015 ) t and y = 1 , 200 e 0.01 t .
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