To solve the problem of how long Carlene must wait to have enough money to buy the desk, we start with the given equation that represents the situation:
[tex]\[ 400 e^{0.06 t} = 500 \][/tex]
Here, [tex]\( 400 \)[/tex] is the initial deposit, [tex]\( 0.06 \)[/tex] is the annual interest rate, [tex]\( t \)[/tex] is the number of years, and [tex]\( 500 \)[/tex] is the desired amount Carlene wants to have.
We'll solve this equation step-by-step for [tex]\( t \)[/tex].
1. Isolate the exponential term:
Divide both sides of the equation by [tex]\( 400 \)[/tex]:
[tex]\[ e^{0.06 t} = \frac{500}{400} \][/tex]
[tex]\[ e^{0.06 t} = 1.25 \][/tex]
2. Apply the natural logarithm ([tex]\( \ln \)[/tex]) to both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ \ln(e^{0.06 t}) = \ln(1.25) \][/tex]
3. Simplify the left side using the property of logarithms that [tex]\( \ln(e^x) = x \)[/tex]:
[tex]\[ 0.06 t = \ln(1.25) \][/tex]
4. Solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(1.25)}{0.06} \][/tex]
Using a calculator to evaluate [tex]\( \ln(1.25) \)[/tex]:
[tex]\[ \ln(1.25) \approx 0.223143551 \][/tex]
Now, substitute this value back into the equation:
[tex]\[ t = \frac{0.223143551}{0.06} \][/tex]
[tex]\[ t \approx 3.719059188 \][/tex]
5. Round to the nearest whole number:
[tex]\[ t \approx 4 \][/tex]
Therefore, Carlene must wait approximately 4 years to have enough money to buy the desk.
The correct answer is 4 years.
