Question

Jacques deposited $1,900 into an account that earns 4% interest compounded semiannually. After t years, Jacques has $3,875.79 in the account. Assuming he made no additional deposits or withdrawals, how long was the money in the account?

Compound interest formula:v(t)=p(1+r/n)^nt

t = years since initial deposit
n = number of times compounded per year
r = annual interest rate (as a decimal)
P = initial (principal) investment
V(t) = value of investment after t years

Answer 1

A year has two semesters, then n = 2\[v(t)=p\left(1+\frac{r}{2}\right)^{2t}\]\[3875.79=1900*\left(1+\frac{0.04}{2}\right)^{2t}\]\[2.0398895=\left(1+\frac{0.04}{2}\right)^{2t}\]Apply natural logarithm on both sides\[ln(2.0398895)=ln\left[\left(1+\frac{0.04}{2}\right)^{2t}\right]\]\[0.712896=2t*ln(1.02)\]\[t=\frac{0.712896}{2*ln(1.02)}\]\[\boxed{\boxed{t=18~years}}\]

Answer 2

thank you!

Answer 3

You're welcome