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Carson invested in an investment account that earns interest compounded continuously. How much money will she have in her account at the end of her th year of investing?
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Q. Carson invested in an investment account that earns interest compounded continuously. How much money will she have in her account at the end of her th year of investing?
- Identify formula for continuous compounding: Identify the formula for continuous compounding.The formula for continuous compounding is given by , where:- is the amount of money accumulated after n years, including interest.- is the principal amount (the initial amount of money).- is the annual interest rate (in decimal form).- is the time the money is invested for in years.- is the base of the natural logarithm, approximately equal to ..
- Plug in values: Plug in the values into the formula.Carson's initial investment () is $\(2000\), the annual interest rate (\( r \)) is \(5\)% or \(0\).\(05\) in decimal form, and the time (\( t \)) is \(8\) years.\(\newline\)So, \( A = 2000 \cdot e^{0.05 \cdot 8} \).
- Calculate exponent part: Calculate the exponent part of the formula.\(\newline\)\( 0.05 \cdot 8 = 0.4 \).\(\newline\)Now, we have \( A = 2000 \cdot e^{0.4} \).
- Calculate value of e^\(0\).\(4\): Calculate the value of \( e^{0.4} \) using a calculator.\(\newline\)\( e^{0.4} \) is approximately \(1\).\(49182\).
- Calculate final amount: Calculate the final amount \( A \).\(\newline\)\( A = 2000 \cdot 1.49182 \).\(\newline\)\( A = 2983.64 \).
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