carlene is saving her money to buy a $500 desk. she deposits $400 into an account with an annual interest…

carlene is saving her money to buy a $500 desk. she deposits $400 into an account with an annual interest rate of 6% compounded continuously. the equation 400e^0.06t = 500 represents the situation, where t is the number of years the money needs to remain in the account. about how long must carlene wait to have enough money to buy the desk? use a calculator and round your answer to the nearest whole number. 4 years 7 years 9 years 20 years

carlene is saving her money to buy a $500 desk. she deposits $400 into an account with an annual interest rate of 6% compounded continuously. the equation 400e^0.06t = 500 represents the situation, where t is the number of years the money needs to remain in the account. about how long must carlene wait to have enough money to buy the desk? use a calculator and round your answer to the nearest whole number. 4 years 7 years 9 years 20 years

Answer

Explanation:

Step1: Isolate the exponential term

Divide both sides of the equation $400e^{0.06t}=500$ by 400. $e^{0.06t}=\frac{500}{400}=\frac{5}{4}$

Step2: Take the natural - logarithm of both sides

$\ln(e^{0.06t})=\ln(\frac{5}{4})$ Since $\ln(e^{x}) = x$, we have $0.06t=\ln(\frac{5}{4})$

Step3: Solve for $t$

$t=\frac{\ln(\frac{5}{4})}{0.06}$ We know that $\ln(\frac{5}{4})\approx\ln(1.25)\approx0.2231$ $t=\frac{0.2231}{0.06}\approx 3.72$

Answer:

4 years