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Sam invests $4,000 in an account that earns 2.5% interest compounded continuously. To determine the amount in the account after a specified amount of time, Sam uses the equation A = P e r t where A is the amount of money in the account after t years, P is the principal, and r is the rate of interest. Sam has a goal of having an account balance of $12,000. Which logarithmic equation can Sam use to determine the number of years it will take to reach his goal? A. t = ln 3 0 . 025 B. t = 3 ln 0 . 025 C. t = 12 , 000 ( ln 4000 0 . 025 ) D. t = 3 ln 0 . 025

Respuesta :

calculista

Answer:

[tex]t=\frac{ln(3)}{0.025}[/tex]

Step-by-step explanation:

we know that

The formula to calculate continuously compounded interest is equal to

[tex]A=P(e)^{rt}[/tex]  

where  

A is the Final Investment Value  

P is the Principal amount of money to be invested  

r is the rate of interest in decimal  

t is Number of Time Periods  

e is the mathematical constant number

we have  

[tex]P=\$4,000\\r=2.5\%=2.5/100=0.025\\A=\$12,000[/tex]  

substitute in the formula above  

[tex]12,000=4,000(e)^{0.025t}[/tex]

[tex]3=(e)^{0.025t}[/tex]  

Apply ln both sides

[tex]ln(3)=ln[(e)^{0.025t}][/tex]  

[tex]ln(3)=(0.025t)ln(e)[/tex]  

Remember that

[tex]ln(e)=1[/tex]

[tex]ln(3)=(0.025t)[/tex]  

[tex]t=\frac{ln(3)}{0.025}[/tex]

[tex]t=43.9\ years[/tex]