how much money should be deposited today in account that earns 8% compounded semiannually so that it will accumulate $8000 in three years

Answer 1

A=P(1+r/n)^nt

where n= number of compounding periods per year

r = annual interest rate= 8%=.08

t = number of years= 3

P= investment

A= investment plus interest= $8000

8000=P(1+.08/2)^2(3)

8000=P(1.04)^6

P = 8000/1.265319018

= $6,322.52

had the compounding been continuous

the formula would have been

A=Pert

8000 = Pe^.08(3)

P = 8000/e^.24

= $6293.02

continuous compounding requires slightly less investment

$29.50 less

using the continously compounding formula is sort of a check

on your answer. If you didn't require slightly more, at biannual compounding, you made a mistake somewhere.