A bank account earns 3 percent interest compounded continuously. What annual amount of money must parents?

each year in order to save 122000 in 14 years for a child's college expenses? Assume the annual amount is added continuously over the period of each year.
A bank account earns 3 percent interest compounded continuously. What annual amount of money must parents?
Your question is unclear. You say to assume the annual amount is added continuously over the year . This does not make sense.





Are you saying that the parents are depositing x$ into a bank account each year starting at year zero and that this is receiving 3% interest compounded continuosly for a year. Then the parents are depositing anther x$ at the beginning of the 2nd year and the continuous compounding continues and this goes on until the beginning of year 14 at which time the parents deposit the final x$ so that at the end of year 14 there will be 122,000 available?





If so, let me know and I will solve it for you.
Reply:edited


i dont learn maths anymore now. so i've forgotten the formula somehow.





122000 %26lt;= x[sum of GP,a=r=1.03,n=14]


122000 %26lt;= x[1.03^15-1.03]/(1.03-1)


122000 %26lt;= 1.03x(1.03^14 -1)/0.03


x %26gt;= 122000*3/103(1.03^14 -1)


x %26gt;= 6932.246





answer=$6932.246 annually, for 14 years.





parents have to put aside $577.69 monthly, or $19.26 per day.
Reply:Let:


the fractional continuous annual interest rate be r per year,


the contribution rate be p per year paid in continuously,


s be the sum accumulated after t years.





ds/dt = p + rs


int (0 to t) ds / (p + rs) = int(0 to t) dt


(1/r)( ln(p + rs) )[0 to s] = t


ln((p + rs) / p) = rt


ln(1 + rs/p) = rt


1 + rs/p = e^(rt)





rs/p = e^(rt) - 1


p = rs / (e^(rt) - 1)


= 0.03 * 122000 / (e^(0.03 * 14) - 1)


= 7012.01 per year.