3. kenny invests $2500 and julia invests $2000. both investments earn 8% annual interest. how much longer…

Explanation:

Step1: Use compound - interest formula

The compound - interest formula is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (as a decimal), and $t$ is the number of years. Given $r=0.08$. For Kenny: $A = 10000$, $P = 2500$. Substitute into the formula: $10000=2500(1 + 0.08)^t_{1}$. First, divide both sides by 2500: $\frac{10000}{2500}=(1.08)^t_{1}$, so $4=(1.08)^t_{1}$. Take the natural logarithm of both sides: $\ln(4)=t_{1}\ln(1.08)$. Then $t_{1}=\frac{\ln(4)}{\ln(1.08)}\approx\frac{1.3863}{0.07696}\approx18.01$ years.

Step2: For Julia

$A = 10000$, $P = 2000$. Substitute into the formula $A = P(1 + r)^t$: $10000=2000(1.08)^t_{2}$. Divide both sides by 2000: $\frac{10000}{2000}=(1.08)^t_{2}$, so $5=(1.08)^t_{2}$. Take the natural logarithm of both sides: $\ln(5)=t_{2}\ln(1.08)$. Then $t_{2}=\frac{\ln(5)}{\ln(1.08)}\approx\frac{1.6094}{0.07696}\approx20.91$ years.

Step3: Calculate the time difference

$\Delta t=t_{2}-t_{1}$. $\Delta t\approx20.91 - 18.01=2.9$ years.

Answer:

Approximately 2.9 years.