$3000 is deposited in an account with a $9\%$ interest rate, compounded continuously. What is the balance after 14 years? $F=\$[?]$ Round to the nearest cent.

Question

GinnyAnswer · Answer

Use the formula for continuous compounding: F = P e r t . 
 Substitute the given values: P = 3000 , r = 0.09 , and t = 14 . 
 Calculate F = 3000 × e 0.09 × 14 = 3000 × e 1.26 . 
 The balance after 14 years is approximately $10576.26 ​ . 
 
 Explanation 
 
 Problem Setup We are given a principal amount of $3000 deposited in an account with an interest rate of 9% compounded continuously. We want to find the balance after 14 years. 
 
 Formula for Continuous Compounding The formula for continuous compounding is given by: F = P e r t where:

F is the future value (balance) 
 P is the principal amount 
 r is the interest rate (as a decimal) 
 t is the time in years

Substituting the Values We are given:

P = $3000 
 r = 9% = 0.09 
 t = 14 years Substituting these values into the formula, we get: F = 3000 × e 0.09 × 14 F = 3000 × e 1.26

Calculating the Future Value Now, we calculate e 1.26 :
 e 1.26 ≈ 3.525421487 So, F = 3000 × 3.525421487 ≈ 10576.26446 
 
 Final Answer Rounding to the nearest cent, we get: F ≈ $10576.26

Examples 
 Continuous compounding is a concept used in finance to calculate the amount of interest earned on an investment over time, assuming the interest is constantly reinvested. For example, if you invest money in a high-yield savings account or a certificate of deposit (CD), the interest earned is often compounded daily or continuously. Understanding continuous compounding can help you estimate the future value of your investments and make informed financial decisions. It's also used in calculating loan payments and other financial instruments where interest accrues over time. For instance, knowing how continuous compounding works can help you compare different investment options and choose the one that offers the best return.

Anonymous · Answer

The balance after 14 years for a $3000 deposit at 9% interest compounded continuously is approximately 10 , 576.26. T hi s i sc a l c u l a t e d u s in g t h e f or m u l a F = Pe^{rt} , w h ere P i s t h e p r in c i p a l , r i s t h e in t eres t r a t e , an d t$ is the time in years. 
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$3000 is deposited in an account with a $9\%$ interest rate, compounded continuously. What is the balance after 14 years? $F=\$[?]$ Round to the nearest cent.

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