Suppose you deposit A dollars into a savings account earning monthly interest R. Through- out the month you withdraw cash at equal intervals and spend it at an even rate so that you have spent all A dollars at the end of the month. Suppose each cash withdrawal has a transaction fee T and that you withdraw cash x times during the month. If you withdraw cash too frequently, you will lose a lot of money to transaction fees. If you withdraw too infrequently, you'll lose money due to interest not accumulating on the cash you have with- drawn. Show that the x that minimizes the money lost to transaction fees and lost interest is given by X = AR 2T Note that the amount of interest that you lose is simple interest and can be calculated I = YR where Y is the average amount of cash you have with you at a given time. Let us make reasonable assumptions on A, R, and T so that 1 ≤ x ≤ 50A.
Suppose you deposit A dollars into a savings account earning monthly interest R. Through- out the month you withdraw cash at equal intervals and spend it at an even rate so that you have spent all A dollars at the end of the month. Suppose each cash withdrawal has a transaction fee T and that you withdraw cash x times during the month. If you withdraw cash too frequently, you will lose a lot of money to transaction fees. If you withdraw too infrequently, you'll lose money due to interest not accumulating on the cash you have with- drawn. Show that the x that minimizes the money lost to transaction fees and lost interest is given by X = AR 2T Note that the amount of interest that you lose is simple interest and can be calculated I = YR where Y is the average amount of cash you have with you at a given time. Let us make reasonable assumptions on A, R, and T so that 1 ≤ x ≤ 50A.
Chapter12: Sequences, Series And Binomial Theorem
Section12.3: Geometric Sequences And Series
Problem 12.58TI: What is the total effect on the economy of a government tax rebate of $500 to each household in...
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Transcribed Image Text:Suppose you deposit A dollars into a savings account earning monthly interest R. Through-
out the month you withdraw cash at equal intervals and spend it at an even rate so that
you have spent all A dollars at the end of the month. Suppose each cash withdrawal has a
transaction fee T and that you withdraw cash x times during the month. If you withdraw
cash too frequently, you will lose a lot of money to transaction fees. If you withdraw too
infrequently, you'll lose money due to interest not accumulating on the cash you have with-
drawn. Show that the x that minimizes the money lost to transaction fees and lost interest
is given by
X =
AR
2T
Note that the amount of interest that you lose is simple interest and can be calculated
I = YR where Y is the average amount of cash you have with you at a given time. Let us
make reasonable assumptions on A, R, and T so that 1 ≤ x ≤ 50A.
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