a loan of $40,000 is made at 3.5% 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 (as a decimal), and $t$ is the number of years. Here, $P=$40000$, $r = 0.035$, and $A\geq57000$. So we have the inequality $57000\leq40000(1 + 0.035)^t$.

Step2: Simplify the inequality

First, divide both sides of the inequality by $40000$: $\frac{57000}{40000}\leq(1.035)^t$, which simplifies to $1.425\leq(1.035)^t$.

Step3: Take the natural logarithm of both sides

$\ln(1.425)\leq\ln(1.035^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $\ln(1.425)\leq t\ln(1.035)$.

Step4: Solve for $t$

$t\geq\frac{\ln(1.425)}{\ln(1.035)}$. Calculate $\frac{\ln(1.425)}{\ln(1.035)}\approx\frac{0.354}{0.0344}\approx10.3$.

Answer:

11