a loan of $32,000 is made at 7.25% interest, compounded annually. after how many years will the amount due…

Explanation:

Step1: Recall 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 (in decimal form), and $t$ is the number of years. Given $P=$32000$, $r = 0.0725$ (since $7.25%=0.0725$), and $A\geq58000$. Substitute these values into the formula: $58000\leq32000(1 + 0.0725)^t$.

Step2: Simplify the inequality

First, divide both sides of the inequality by $32000$: $\frac{58000}{32000}\leq(1.0725)^t$. $\frac{29}{16}\leq(1.0725)^t$.

Step3: Take the natural logarithm of both sides

$\ln(\frac{29}{16})\leq\ln(1.0725^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $\ln(\frac{29}{16})\leq t\ln(1.0725)$.

Step4: Solve for $t$

$t\geq\frac{\ln(\frac{29}{16})}{\ln(1.0725)}$. We know that $\ln(\frac{29}{16})=\ln(29)-\ln(16)\approx3.3673 - 2.7726=0.5947$ and $\ln(1.0725)\approx0.0699$. So $t\geq\frac{0.5947}{0.0699}\approx8.51$.

Answer:

9 years