Svetlana won $1,000,000 in a contest, to be paid in twenty $50,000 payments at yearly intervals, the rst payment paid at the time of the contest. (of course, the present value of her winningis less than $1,000,000.) svetlana decided to keep x each year to spend and deposit theremaining $50,000-x into an account earning an annual eective interest rate of 5%. shechose the value x to be as large as possible so that, at the moment of the 20th deposit, theaccount would have grown to such a size that it would provide svetlana and her heirs at leastx per year in interest forever. find x.

The amount X she can retire out of every payment is $31,155.52.

Step-by-step explanation:

We start calculating the amount needed to provide an perpetuity of X after the 20th deposit. The formula for the the present value of a perpetuity is

This is the amount of capital she has to have in her account to provide X yearly forever.

We have 20 payments, which are compunded at an effective rate of 5%. The amount deposit out every payment is (50,000-X).

We can write a timeline to see all the deposits

We can express then the capitalization of the deposit (C) as

This capital C has to be equal to 20X:

Then, the amount X she can retire out of every payment is $31,155.52.

step-by-step explanation:

the triangles are similar. therefore the corresponding sides are in proportion:

              cross multiply

            divide both sides by 32