Question

1. allison invested $23,000 in an account paying an interest rate of 6.7% compounded annually. assuming no deposits or withdrawals are made, how long would it take, to the nearest year, for the value of the account to reach $76,300? p=23000

Explanation:

Step1: Recall compound interest formula

The formula for compound interest compounded annually 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), and $t$ is the time in years.
Given $P = 23000$, $r = 0.067$ (since $6.7\%=0.067$), and $A = 76300$. Substitute these values into the formula:
$$76300 = 23000(1 + 0.067)^t$$

Step2: Simplify the equation

Divide both sides by $23000$:
$$\frac{76300}{23000}=(1.067)^t$$
Simplify $\frac{76300}{23000}\approx3.3174$:
$$3.3174=(1.067)^t$$

Step3: Take natural logarithm on both sides

Take $\ln$ of both sides:
$$\ln(3.3174)=\ln((1.067)^t)$$
Using the logarithm power rule $\ln(a^b)=b\ln(a)$:
$$\ln(3.3174)=t\ln(1.067)$$

Step4: Solve for $t$

Divide both sides by $\ln(1.067)$:
$$t = \frac{\ln(3.3174)}{\ln(1.067)}$$
Calculate $\ln(3.3174)\approx1.198$ and $\ln(1.067)\approx0.0649$:
$$t=\frac{1.198}{0.0649}\approx18.46$$

Step5: Round to nearest year

Round $18.46$ to the nearest year, which is $18$.

Answer:

18