An annuity: Suppose you are able to find an investment that pays a monthly interest rate of r as a decimal. You want to invest P dollars that will help support your child. If you want your child to be...

An annuity: Suppose you are able to find an investment that pays a monthly interest rate of r as a decimal. You want to invest P dollars that will help support your child. If you want your child to be able to withdraw M dollars per month for t months, then the amount you must invest is given by

A fund such as this is known as an annuity. For the remainder of this problem, we suppose that you have found an investment with a monthly interest rate of 0.01 and that you want your child to be able to withdraw $200 from the account each month.

a. Find a formula for your initial investment P as a function of t, the number of monthly withdrawals you want to provide, and make a graph of P versus t. Be sure your graph shows up through 40 years (480 months).

b. Use the graph to find out how much you need to invest so that your child can withdraw $200 per month for 4 years.

c. How much would you have to invest if you wanted your child to be able to withdraw $200 per month for 10 years?

d. A perpetuity is an annuity that allows for withdrawals for an indefinite period. How much money would you need to invest so that your descendants could withdraw $200 per month from the account forever? Be sure to explain how you got your answer.