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Question 84045: Can anyone help me with this problem? I'm in an class and running on a dealine. I'm not very capable when it comes to this stuff so any help I can get will be appreciated. Thanks.
The formula for calculating the amount of money returned for deposit money into a bank account or CD (Certificate of Deposit) is given by the following:
A is the amount of returned
P is the principal amount deposited
r is the annual interest rate (expressed as a decimal)
n is the compound period
t is the number of years
Carry all calculations to 6 decimals on all assignments then round the answer to the
nearest cent.
Suppose you deposit $20,000 for 3 years at a rate of 8%.
a) Calculate the return (A) if the bank compounds annually (n = 1).
Answer:
Show work in this space. Use ^ to indicate the power.
b) Calculate the return (A) if the bank compounds quarterly (n = 4). Round your answer to the hundredth's place.
Answer:
Show work in this space .
c) Calculate the return (A) if the bank compounds monthly (n = 12). Round your answer to the hundredth's place.
Answer:
Show work in this space.
d) Calculate the return (A) if the bank compounds daily (n = 365). Round your answer to the hundredth's place.
Answer:
Show work in this space.
e) What observation can you make about the size of increase in your return as your compounding increases more frequently?
Answer:
f) If a bank compounds continuous, then the formula becomes simpler, that is where e is a constant and equals approximately 2.7183. Calculate A with continuous compounding. Round your answer to the hundredth's place.
Answer:
g) Now suppose, instead of knowing t, we know that the bank returned to us $25,000 with the bank compounding continuously. Using logarithms, find how long we left the money in the bank (find t). Round your answer to the hundredth's place.
Answer:
Show work here:
h) A commonly asked question is, “How long will it take to double my money?” At 8% interest rate and continuous compounding, what is the answer? Round your answer to the hundredth's place.
Answer:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!  Start with the given equation
 Plug in p=20000, r=0.08
a)
Lets calculate the return if the bank compounds annually
Let n=1 and plug it into
 Start with the given expression
 Divide 0.08 by 1 to get 0.08
 Multiply the exponents 1 and 3 to get 3
 Add 1 and 0.08 to get 1.08
 Raise 1.08 to 3 to get 1.259712
 Multiply 20000 and 1.259712 to get 25194.24
So our return is $25194.24
b)
Lets calculate the return if the bank compounds quarterly
Let n=4 and plug it into
 Start with the given expression
 Divide 0.08 by 4 to get 0.02
 Multiply the exponents 4 and 3 to get 12
 Add 1 and 0.02 to get 1.02
 Raise 1.02 to 12 to get 1.26824179456255
 Multiply 20000 and 1.26824179456255 to get 25364.8358912509
So our return is $25364.84
c)
Lets calculate the return if the bank compounds monthly
Let n=12 and plug it into
 Start with the given expression
 Divide 0.08 by 12 to get 0.00666666666666667
 Multiply the exponents 12 and 3 to get 36
 Add 1 and 0.00666666666666667 to get 1.00666666666667
 Raise 1.00666666666667 to 36 to get 1.27023705162065
 Multiply 20000 and 1.27023705162065 to get 25404.741032413
So our return is $25404.74
d)
Lets calculate the return if the bank compounds daily
Let n=365 and plug it into
 Start with the given expression
 Divide 0.08 by 365 to get 0.000219178082191781
 Multiply the exponents 365 and 3 to get 1095
 Add 1 and 0.000219178082191781 to get 1.00021917808219
 Raise 1.00021917808219 to 1095 to get 1.27121572005174
 Multiply 20000 and 1.27121572005174 to get 25424.3144010349
So our return is $25424.31
e) What observation can you make about the size of the increase in your return as your compounding increases more frequently?
As the compounding frequency increases, the return slowly approaches some finite number (which in this case appears to be about $12213.69). Think about it, banks wouldn't be too fond of shelling out an infinite amount of cash.
f)Calculate A with continuous compounding
Using the contiuous compounding formula  where e is the constant 2.7183 and letting r=0.1, P=10,000, and t=2 we get
 Start with the given equation
 Multiply 0.1 and 2
 Raise 2.7183 to 0.2
 Multiply
So using continuous compounding interest we get a return of $25425.02 (which is real close to what we got from a daily compounding frequency)
g)Now suppose, instead of knowing t, we know that the bank returned to us $15,000 with the bank compounding continuously. Using natural logarithms, find how long we left the money in the bank (find t)
 Divide both sides by 20,000
 Take the natural log of both sides. This eliminates "e".The natural log (pronounced "el" "n") is denoted "ln" on calculators.
 Divide both sides by 0.08
So we get
So it will take a little over 2 and a half years to generate $25,000
h) A commonly asked question is, “How long will it take to double my money?” At 8% interest rate and continuous compounding, what is the answer?
Since we want to double our money, let A=2*20,000. So A=40,000. Now solve for t:
 Divide both sides by 10,000
 Take the natural log of both sides. This eliminates "e".The natural log (pronounced "el" "n") is denoted "ln" on calculators.
 Divide both sides by 0.08
So we get
So it will take about 8 and a half years to double your money.
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