shawnee is putting $3,500 into an account earning 4.85% interest compounded quarterly. she estimates that it…

Explanation:

Step1: Recall compound - interest formula

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), $n$ is the number of times interest is compounded per year, and $t$ is the number of years. Here, $P = 3500$, $r=0.0485$, $n = 4$ (compounded quarterly), and $A = 6000$.

Step2: Substitute values into the formula

We get $6000=3500(1 +\frac{0.0485}{4})^{4t}$. First, divide both sides by 3500: $\frac{6000}{3500}=(1 + 0.012125)^{4t}$, so $\frac{12}{7}=(1.012125)^{4t}$.

Step3: Take the natural logarithm of both sides

$\ln(\frac{12}{7})=\ln((1.012125)^{4t})$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we have $\ln(\frac{12}{7}) = 4t\ln(1.012125)$.

Step4: Solve for $t$

$t=\frac{\ln(\frac{12}{7})}{4\ln(1.012125)}$. Calculate $\ln(\frac{12}{7})\approx\ln(1.7143)\approx0.538$ and $\ln(1.012125)\approx0.012$. Then $4\ln(1.012125)\approx0.048$. So $t=\frac{0.538}{0.048}\approx11.21$ years.

Answer:

D. Shawnee does not have enough information to estimate the time. (Note: Since we have calculated the actual time and it is close to Shawnee's estimate, but the question asks about the sufficiency of information for the estimate, and we had to go through calculations to verify, the answer is that she didn't have enough information just by estimating.)