Jeff invested $[tex]$3000$[/tex] in an account that earns [tex]$6.5 %$[/tex] interest, compounded annually. The formula for compound interest is [tex]$A(t)=P(1+i)^t$[/tex].
How much did Jeff have in the account after 3 years?
A. $[tex]$9585.00$[/tex]
B. $[tex]$3390.00$[/tex]
C. $[tex]$13,476.38$[/tex]
D. $[tex]$3623,85$[/tex]
Answer (2)
Recall the compound interest formula: A ( t ) = P ( 1 + i ) t .
Substitute the given values: P = $3000 , i = 0.065 , and t = 3 .
Calculate A ( 3 ) = 3000 ( 1 + 0.065 ) 3 = 3000 ( 1.065 ) 3 ≈ $3623.85 .
The amount Jeff had in the account after 3 years is $3623.85 .
Explanation
Understanding the Problem We are given the principal amount, the interest rate, and the time period. We need to find the amount in the account after 3 years using the compound interest formula.
Recalling the Formula The formula for compound interest is given by A ( t ) = P ( 1 + i ) t , where:
A ( t ) is the amount after time t
P is the principal amount
i is the interest rate
t is the time in years
Identifying Given Values We are given:
P = $3000
i = 6.5% = 0.065
t = 3 years
Substituting Values Substitute the given values into the formula: A ( 3 ) = 3000 ( 1 + 0.065 ) 3
Calculating the Final Amount Calculate the amount: A ( 3 ) = 3000 ( 1.065 ) 3 A ( 3 ) = 3000 × 1.207946125 A ( 3 ) = 3623.838375 Rounding to two decimal places, we get: A ( 3 ) = $3623.84 or $3623.85
Selecting the Correct Answer Comparing the calculated value with the given options, we find that option D is the closest to our calculated amount.
Final Answer Therefore, the amount Jeff had in the account after 3 years is approximately $3623.85 .
Examples
Compound interest is a powerful tool for growing wealth over time. For example, if you invest money in a retirement account that earns compound interest, your investment will grow faster than if it earned simple interest. Understanding compound interest can help you make informed decisions about your savings and investments. Let's say you invest $1000 in an account that earns 5% interest, compounded annually. After 10 years, your investment will grow to approximately $1628.89 . This demonstrates the long-term benefits of compound interest.
After calculating the compound interest for Jeff's investment of $3000 at a rate of 6.5% compounded annually for 3 years, we find that he has approximately $3623.85 in his account. The correct answer is option D. This amount is achieved by applying the formula A(t) = P(1+i)^t and substituting the appropriate values.
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