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Question 73568: Could you please help me solve this interest word problem? I'm clueless and don't know where to start! Here is the 2 part question.
The formula for calculating the amount of money returned for an initial deposit money into a bank account or CD is given by
A=P( 1+r)^nt
-
n
However, my question is 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 have left the money in the bank (find t). Round your answer to the hundredth's place.
Found 2 solutions by stanbon, bucky:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! we know that the bank returned to us $15,000 with the bank compounding continuously. Using natural logarithms, find how long we have left the money in the bank (find t). Round your answer to the hundredth's place.
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If it was compounded continuously the formla you want is:
A= Pe^rt
Then
15000=Pe^rt
COMMENT: You did not post P, the initial investment, or r, the annual
rate of interest. Without them you cannot get a number figure for "t".
You can get a symbolic answer.
e^rt = 15000/P
Take the natural log to get:
rt = ln(15000/P)
Then:
t= (1/r)ln(15000/P)
=============
Cheers
Stan H.
Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website! Given:
.
.
where A is the amount returned, P is the initial deposit, n is the number of times the compounding
takes place in a year, r is the annual interest rate expressed as a decimal, and t is the
number of years the investment remains on deposit.
.
Actually, for continuous compounding, which is called for in the problem, the formula
that should be used is:
.
.
where A, P, r, and t are as defined above and e is the base of the natural logarithms.
.
You picked the wrong equation for continuous compounding. The one you gave is more useful
for a limited number of compoundings in a year ... twice a year, quarterly, monthly or
weekly.
.
So you need to proceed with the correct equation to use in solving for t subject to continuous
compounding:
.
.
Now substitute $15,000 for A to get:
.
.
the fact that there is an exponent that contains the unknown suggests that we take the
logarithm of both sides so we can use the logarithmic property that in a logarithm
of a quantity having an exponent you can remove the exponent and use it as a multiplier
of the logarithm. And since e (the base of natural logs) is involved in the equation,
this suggests that we take the natural log of both sides of this equation (ln is the way
we identify natural logarithms).
.
I haven't figured a way to make the formula translator at this site work on natural logs.
Therefore, I'm going back to the text way of writing equations. Sorry ...
After taking the natural logarithm of both sides we have:
.
ln(15000) = ln(P*e^(rt))
.
A calculator tells you that ln(15000) = 9.61580548 Substitute this to give:
.
9.61580548 = ln(P*e^(rt))
.
By the rules of logarithms the log of a product is equal to the sums of the logs of the
terms in the product. Therefore we can write the right side of this equation as:
.
9.61580548 = ln(P) + ln(e^(r*t))
.
and as explained above, the exponent (rt) can be written as the multiplier of the log
to give us:
.
9.61580548 = ln(P) + (r*t)*ln(e)
.
But ln(e) equals 1 and this reduces the problem to:
.
9.61580548 = ln(P) + r*t
.
ln(P) is just a number you can get from a calculator. In solving for t you need it on
the other side of the equation, so subtract ln(P) from both sides of the equation to
give you:
.
9.61580548 - ln(P) = r*t
.
Now you can solve for t (in years) by dividing the entire left side of this equation by
r. I would help you with this, but you didn't provide P or r in your description of
the problem. (r should be something like .06 for a 6% annual interest rate. Use the
decimal form).
.
Hope this helps you to get to a solution for this problem.
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