`r = 6%, t = 40` Find the principal P that must be invested at a rate r, compounded monthly, so that $1,000,000 will be available for...

The formula in compounding interest is

`A = P(1 + r/n)^(n*t)`

where

A is the accumulated amount

P is the principal

r is the annual rate

n is the number of compounding periods in a year, and

t is the number of years.

Plugging in the values A = 1000000, r = 0.06 and t = 40 , the formula becomes:

`1000000=P(1+0.06/n)^(n*40)`

Since the r is compounded monthly, the value of n is 12.

`1000000=P(1+0.06/12)^(12*40)`

Simplifying the right side, it becomes

`1000000=P(1+0.005)^480`

`1000000=P(1.005)^480`

Isolating the P, it yields

`1000000/1.005^480=(P(1.005)^480)/1.005^480`

`91262.08=P`

Therefore, the principal amount that should be invested is $91,262.08 .