Chang's $2000 investment at a 9.8% annual interest rate, compounded semiannually, will take approximately 4.42 years to grow to $3076 when rounded to the nearest hundredth.
Chang deposited $2000 into an account with a 9.8% annual interest rate, compounded semiannually. To calculate how long it will take for this investment to grow to $3076, we can use the formula for compound interest:
A = P(1 + r/n)(nt)
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (in decimal form).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested or borrowed for, in years.
We can rearrange the formula to solve for t (time), the unknown variable in this scenario:
t = (log(A/P)) / (n × log(1 + r/n))
By substituting the values into the formula, we get:
t = (log(3076/2000)) / (2 × log(1 + 0.098/2))
t = (log(1.538)) / (2 ×log(1.049))
t = (0.1872) / (2 ×0.0212)
t = 0.1872 / 0.0424
t = 4.4151 years
So, it would take approximately 4.42 years for Chang's investment to grow to $3076 when rounded to the nearest hundredth.