Question

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

Answer 1

`t=(ln(3))/(0.025)`

Explanation:

we know that

The formula to calculate continuously compounded interest is equal to

`A=P(e)^(rt)`

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

`P=\$4,000\\r=2.5\%=2.5/100=0.025\\A=\$12,000`

substitute in the formula above

`12,000=4,000(e)^(0.025t)`

`3=(e)^(0.025t)`

Apply ln both sides

`ln(3)=ln[(e)^(0.025t)]`

`ln(3)=(0.025t)ln(e)`

Remember that

`ln(e)=1`

`ln(3)=(0.025t)`

`t=(ln(3))/(0.025)`

`t=43.9\ years`