Rudy wants to invest $\$ 1000$ in savings certificates that bear an interest rate of $7.5 \%$ compounded semiannually. How long will it take for the amount to be $\$ 1500 ?$
A) $2.54$ yrs
B) $3.08$ yrs
C) $5.51$ yrs
D) $9.25$ yrs
E) $10.36$ yrs

Question

Quizplus · Accepted Answer

To solve this problem, we use the formula for compound interest: $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the amount of money accumulated after n years, including interest, $P$ is the principal amount (\$1000), $r$ is the annual interest rate (7.5%, or 0.075 as a decimal), $n$ is the number of times that interest is compounded per year (2, since it's semiannually), and $t$ is the time the money is invested for in years.We are solving for $t$ when $A = \$1500$, $P = \$1000$, $r = 0.075$, and $n = 2$.Rearranging the formula to solve for $t$, we get:$$1500 = 1000(1 + \frac{0.075}{2})^{2t}$$$$1.5 = (1 + \frac{0.075}{2})^{2t}$$$$1.5 = (1.0375)^{2t}$$Taking the natural logarithm (ln) of both sides:$$\ln(1.5) = \ln((1.0375)^{2t})$$$$\ln(1.5) = 2t \ln(1.0375)$$$$t = \frac{\ln(1.5)}{2 \ln(1.0375)}$$Plugging in the values:$$t \approx \frac{0.4054651}{2 \cdot 0.0368142} \approx 5.51$$Therefore, it will take approximately 5.51 years for the amount to grow to $1500.

Rudy Wants to Invest $1000\$ 1000$1000 In Savings Certificates That Bear an Interest Rate Of

Rudy Wants to Invest $\$ 1000$ In Savings Certificates That Bear an Interest Rate Of