Question

jack deposited $2,902.00 into a new savings account that earns 5% interest compounded continuously. how long will it take for the balance to grow to $3,223.00? round your answer to the nearest month. years and months submit

Explanation:

Step1: Recall the formula for continuous compounding

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal), and $t$ is the time in years.
We know that $P = 2902$, $A = 3223$, and $r = 0.05$ (since 5% = 0.05). We need to solve for $t$.

Step2: Substitute the known values into the formula

Substitute $P$, $A$, and $r$ into the formula:
$$3223 = 2902e^{0.05t}$$

Step3: Divide both sides by 2902

$$\frac{3223}{2902}=e^{0.05t}$$
Calculate $\frac{3223}{2902}\approx1.1106$. So we have:
$$1.1106 = e^{0.05t}$$

Step4: Take the natural logarithm of both sides

Take the natural log of both sides: $\ln(1.1106)=\ln(e^{0.05t})$.
Since $\ln(e^{x}) = x$, this simplifies to:
$$\ln(1.1106)=0.05t$$

Step5: Solve for $t$

We know that $\ln(1.1106)\approx0.1053$. So:
$$t=\frac{\ln(1.1106)}{0.05}\approx\frac{0.1053}{0.05} = 2.106\text{ years}$$

Step6: Convert the decimal part of $t$ to months

The decimal part of $t$ is $0.106$ years. To convert this to months, multiply by 12 (since there are 12 months in a year):
$$0.106\times12\approx1.272\text{ months}$$
Rounding to the nearest month, the decimal part is approximately 1 month. The whole number part of $t$ is 2 years.

Answer:

2 years and 1 month