Question 34491This question is from textbook COLLEGE ALGEBRA WITH MODELING AND VISUALIZATION
: 3) The formula for calculating the amount of money returned for an initial deposit money into a bank account or CD (Certificate of Deposit) is given by
A is the amount of returned.
P is the principal amount initially deposited.
r is the annual interest rate (expressed as a decimal).
n is the compound period.
t is the number of years.
Suppose you deposit $10,000 for 2 years at a rate of 10%.
a) Calculate the return (A) if the bank compounds annually (n = 1).
Answer:
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b) Calculate the return (A) if the bank compounds quarterly (n = 4).
Answer:
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c) Calculate the return (A) if the bank compounds monthly (n = 12).
Answer:
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d) Calculate the return (A) if the bank compounds daily (n = 365).
Answer:
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e) What observation can you make about the increase in your return as your compounding increases more frequently?
Answer:
f) If a bank compounds continuous, then the formula takes a simpler, that is
where e is a constant and equals approximately 2.7183.
Calculate A with continuous compounding.
Answer:
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g) Now suppose, instead of knowing t, we know that the bank returned to us $15,000 with the bank compounding continuously. Using logarithms, find how long we left the money in the bank (find t).
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h) A commonly asked question is, “How long will it take to double my money?” At 10% interest rate and continuous compounding, what is the answer?
Answer:
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4) For a fixed rate, a fixed principal amount, and a fixed compounding cycle, the return is an exponential function of time. Using the formula, , let r = 10%, P = 1, and n = 1 and give the coordinates (t, A) for the points where t = 0, 1, 2, 3, 4.
a) Show coordinates in this space
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b) Show graph here
This question is from textbook COLLEGE ALGEBRA WITH MODELING AND VISUALIZATION
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! I'll do a few of these as examples.
#b A=P(1+r/n)^(nt)
A=10000(1+0.10/4)^[(4)(2)]
A=10000(1+0.025)^8
A=10000(1.025)^8
A=$12184.03
#f A=Pe^(rt)
A=10000e^(0.10(2))
A=10000e^0.20
A=$12214.03
#g 15000=10000e^(0.10t)
1.5 =e^0.10t
Take the natural log of both sides to get:
ln1.5==0.1t
0.40465108...=0.1t
t=4.05 years
#h 20,000=10000e^(0.1t)
2=e^0.1t
Take the natural log of both sides to get:
0.69=0.1t
t=6.9 years (This is called "the rule of seven".)
Cheers,
Stan H.
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