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.\n4 years\n7 years\n9 years\n20 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.\n4 years\n7 years\n9 years\n20 years

Answer

Explanation:

Step1: Isolate the exponential term

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

Step2: Take the natural - logarithm of both sides

Since $y = e^{x}$ and $y=\ln(x)$ are inverse functions, taking the natural - logarithm of both sides of $e^{0.06t}=\frac{5}{4}$ gives $\ln(e^{0.06t})=\ln(\frac{5}{4})$. By the property $\ln(e^{x}) = x$, the left - hand side simplifies to $0.06t$. So, $0.06t=\ln(\frac{5}{4})$.

Step3: Solve for $t$

We know that $\ln(\frac{5}{4})\approx0.2231$. Then $t=\frac{\ln(\frac{5}{4})}{0.06}=\frac{0.2231}{0.06}\approx3.72$.

Step4: Round to the nearest whole number

Rounding 3.72 to the nearest whole number gives $t = 4$.

Answer:

A. 4 years