Question

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 calculator and round your answer to the nearest whole number. 4 years 7 years 9 years 20 years

Explanation:

Step1: Solve the equation for \(t\)

Given \(400e^{0.06t}=500\), first divide both sides by \(400\):
\(e^{0.06t}=\frac{500}{400} = 1.25\)

Step2: Take the natural logarithm of both sides

Since \(y = e^{x}\) and \(x=\ln(y)\) are inverse functions, \(\ln(e^{0.06t})=\ln(1.25)\)
Using the property \(\ln(e^{a})=a\), we get \(0.06t=\ln(1.25)\)

Step3: Calculate \(\ln(1.25)\) and solve for \(t\)

We know that \(\ln(1.25)\approx0.2231\)
Then \(t=\frac{\ln(1.25)}{0.06}=\frac{0.2231}{0.06}\approx 3.72\approx4\)

Answer:

4 years