Question

suppose that $11,000 is invested in a bond fund and the account grows to $14,243.93 in 5 yr.
part 1 of 2
(a) use the model $a = pe^{rt}$ to determine the average rate of return under continuous compounding. round to the nearest tenth of a percent. avoid rounding in intermediate steps.
the average rate of return under continuous compounding is approximately 5.2%.
part: 1 / 2
part 2 of 2
(b) how long will it take the investment to reach $19,000 if the rate of return continues? round to the nearest tenth of a year. round values in intermediate steps to three decimal places.
it will take the investment approximately yr to reach $19,000 if the rate of return continues.

Explanation:

Step1: Recall the formula

We use the formula $A = Pe^{rt}$, where $A=\$19000$, $P = \$11000$, and $r=0.052$.

Step2: Substitute values into the formula

Substitute into $A = Pe^{rt}$: $19000=11000e^{0.052t}$.

Step3: Solve for $t$

First, divide both sides by $11000$: $\frac{19000}{11000}=e^{0.052t}$, so $\frac{19}{11}=e^{0.052t}$.
Then, take the natural logarithm of both sides: $\ln(\frac{19}{11})=\ln(e^{0.052t})$.
Since $\ln(e^{x}) = x$, we have $\ln(\frac{19}{11}) = 0.052t$.
Now, calculate $\ln(\frac{19}{11})\approx\ln(1.727)\approx0.546$.
So, $t=\frac{0.546}{0.052}\approx10.5$. Rounding to the nearest tenth gives $t\approx10.0$.

Answer:

It will take the investment approximately $10.0$ yr to reach $\$19,000$ if the rate of return continues.