Question

find the value of $1,300 invested at 4.2% interest compounded continuously for 5 years,9 months. round your answer to the nearest hundreth or cent. use e=2.718

Answer 1

please help me i really dont get this :(

Answer 2

Compounded continuously? I don't know any way to explain that except with calculus. The amount of interest dM awarded in a tiny sliver of time dt is:
\[dM = M \frac{i}{1/dt}\]
Where i is the interest rate per year (4.2% or 0.042) and dt is measured in years. The 1 in the denominator is because to get the amount of interest we should award over dt years, we need to divide the simple interest rate by 1/dt. For example, if dt = 0.5 (six months), then we need to divide the annual simple interest rate of 4.2% by 1/0.5 = 2, to get 2.1%, because we should give 2.1% interest over six months.

If, on the other hand, dt = 1/365 (one day), then we need to divide the annual simple interest rate by 1/dt = 365 to get 0.0115...%, which is the amount of interest we should award in one day. To get the interest we should award over 1 hour, we use dt = 1/(365*24) = 0.000114... and so forth.

Now simplify the equation:
\[\frac{dM}{M} = i dt\]
Integrate both sides:
\[ln M = i t + C\]
\[M = M(0) e^{i t}\]
Substitute in your values:
\[M = $1300 e^{(0.042)*5.75)} = $1655.10.

Answer 3

Oops, last line should read:
\[M = $1300 e^{(0.042)(5.75)} = $1655.10\]

Answer 4

Also note that 5 years, 9 months is 5.75 years.

Answer 5

oh wow thankyou so much! i have one more if you dont mind? sorry im new to this site i dont know how to work it really also i wanna give you a medal