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A woman invests $6300 in an account that pays 6% interest per year, compounded continuously.(a) What is the amount after 2 years? (Round your answer to the nearest cent.)$ (b) How long will it take for the amount to be $8000? (Round your answer to two decimal places.) yr

A Woman Invests 6300 In An Account That Pays 6 Interest Per Year Compounded Continuouslya What Is The Amount After 2 Years Round Your Answer To The Nearest Cent class=

Sagot :

Given: A woman invests $6300 in an account that pays 6% interest per year, compounded continuously.

Required: a) To determine the amount after 2 years.

b) To determine how long it will take for the amount to be $8000.

Explanation: The amount, A after t years with an interest rate of r is given by-

[tex]A=Pe^{rt}[/tex]

Here,

[tex]\begin{gathered} P=6300 \\ r=\frac{6}{100} \\ =0.06 \\ t=2 \end{gathered}[/tex]

Substituting the values, we get-

[tex]\begin{gathered} A=6300e^{0.06\times2} \\ =7103.23 \end{gathered}[/tex]

Hence the amount after 2 years is $7103.23

Next, let t be the time it takes for the amount to be $8000-

[tex]\begin{gathered} 8000=6300e^{0.06t} \\ \frac{8000}{6300}=e^{0.06t} \\ \ln(1.2698)=0.06t \end{gathered}[/tex]

Further solving for t as-

[tex]t=3.98\text{ years}[/tex]

Hence, it takes 3.98 years for the amount to be $8000.

Final Answer: a) $7103.23

b) 3.98 years

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