`r = 7.5% , t = 20` 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 given values, the formula becomes:

`1000000=P(1+0.075/n)^(n*20)`

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

`1000000=P(1+0.075/12)^(12*20)`

Simplifying the right side, it becomes

`1000000=P(1+0.00625)^240`

`1000000=P(1.00625)^240`

Isolating the P, it yields

`1000000/1.00625^240=(P(1.00625)^240)/1.00625^240`

`224174.18=P`

Therefore, the principal amount that should be invested is $224,174.18 .