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continuously compounded interest (solving for r)?
Hi there ^_^,
This came from a long question in my book, but I think this should be enough info.
At the end of 100 years from the reception of Franklin's gift, in Jan. 1894, the fund had grown from 1000 pounds to almost exactly 90,000 pounds. In 100 years the original capital had multiplied about 90 times instead of the 131 times Franklin had imagined.
What rate of interest, compounded continuously for 100 years, would have multiplied Franklin's original capital by 90?
Now, I know the formula for this is A(t) = A_0 * e^rt.
I came up with:
A(100) = 90,000 * e^r100
-90,000 = e^r100
Now, if I did this, I could not use ln to cancel out e because of the -90,000. *sighs* I do not even know if I am using the right number for A_0. Could someone help me.
I thank God for your reply and help ^_^
~beauty7 ^_^
btw, I also know that the answer is 4.50% but I do not know how to get there. :(
I appreciate all of your answers. ^_^ They were all very helpful.^_^ By the way, buying a financial calculator would be nice Danny S but it would not help me on my test though. ^_^ Again, I thank God for all of your answers. ^_^
6 Answers
- mathmaniacLv 61 decade agoFavorite Answer
The formula is correct, but your working is all wrong.
A_0 = initial amount, i.e. 1000 pounds.
r = the interest rate of interest (don't mind the pun)
t = number of years, i.e. 100
A(t) = final amount after t years
Therefore the working is as follows:
90,000 = 1000 * e^100r
90 = e^100r
ln 90 = 100r
r = (ln 90)/100 = 4.5% approx
- Jun AgrudaLv 71 decade ago
= 1,000.00([1 + {0.0450825701203947/12}]^[100 * 12])
= 1,000.00([1 + {0.003756880843366}]^[1,200])
= 1,000.00([1.003756880843366]^[1,200])
= 1,000.00(90)
= 90,000.00
The answer is in the 1st line of the foregoing and develops after each line simplifies the figures. The rate therefore is not exactly 4.5% but 4.50825701203947%. You can prove this by implementing the formula in either Microsoft Excel spreadsheet or code it in a Visual Basic 6.0 (VB 6.0) if you know some programming techniques. It would be very cumbersome to do the computation with the use of calculators.
- sahsjingLv 71 decade ago
Use your formula, A(t) = A(0) * e^rt.
A(100) = 90(1,000) = 90,000,
A(0) = 1,000
t = 100
90,000 = 1,000*e^100t
Solve for t,
t = (1/100)ln90 = 4.50%
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- 1 decade ago
The formula is set up as
final amount = initial amount * e ^ ((rate)(time))
90,000=1000e^(100r)
90=e^(100r)
ln 90 = 100r
(ln 90)/100=r
r=0.045 * (100%) = 4.50%
Source(s): Calc
