a loan of $18,000 is made at 4.25% interest, compounded annually. after how many years will the amount due…

Answer:

13

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. Here, $P=$18000$, $r = 0.0425$, and $A\geq31000$. So we have the inequality $31000\leq18000(1 + 0.0425)^t$.

Step2: Simplify the inequality

Divide both sides of the inequality by 18000: $\frac{31000}{18000}\leq(1.0425)^t$, which simplifies to $\frac{31}{18}\leq(1.0425)^t$.

Step3: Take the natural logarithm of both sides

$\ln(\frac{31}{18})\leq\ln(1.0425^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $\ln(\frac{31}{18})\leq t\ln(1.0425)$.

Step4: Solve for $t$

$t\geq\frac{\ln(\frac{31}{18})}{\ln(1.0425)}$. Calculate $\ln(\frac{31}{18})\approx\ln(31)-\ln(18)\approx3.43399 - 2.89037=0.54362$ and $\ln(1.0425)\approx0.0416$. Then $t\geq\frac{0.54362}{0.0416}\approx13.0678$. Since $t$ must be a whole number, the smallest possible whole - number value of $t$ is 13.