Question 383490: Please help me solve this word problem: Formulas: y=be^rt; A=P(1+(r/n)^nt
George has $65 to invest. a. The bank gives him 8.2% continuous interest, how long will it take for George to accrue $100? b. At 7.6% interest compound monthly, how much money will George have in 12 years?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! y = b * e^rt
sounds like this is the continuous compounding formula.
george has $65 to invest.
at 8.2% continuous interest, how long will it take for george to accrue $100?
y = $100
b = $65
r = .082
t = time in years
formula is $100 = $65 * e^(.082*t)
divide both sides of this formula by 65 to get:
100/65 = e^(.082*t)
take log of both sides to get:
log(100/65) = log(e^(.082*t))
by laws of logarithms, log(x^y) = y*log(x).
formula becomes log(100/65) = .082*t * log(e)
divide both sides of equation by log(e) to get:
log(100/65) / log(e) = .082*t
divide both sides of this equation by .082 to get:
t = (log(100/65) / log(e)) / .082
log(100/65) = .187086643
log(e) = log(2.718281828) = .434294482
log(100/65) / log(e) = .430782916
.430782916 / .082 = 5.253450196
you get t = (log(100/65) / log(e)) / .082 = 5.253450196.
If we did this right, then t = 5.253450196 years
your original formula is:
100 = 65 * e^(.082*t)
this becomes:
100 = 65 * e^(.082*5.253450196)
e represents the scientific constant of 2.718281828
100 = 65 * e^(.082*5.253450196) becomes:
100 = 65 * 2.718182818^(.082*5.253450196).
Use your calculator to see that 100 = 100, making t = 5.253450196 correct.
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second problem.
At 7.6% interest compound monthly, how much money will George have in 12 years?
Formula they show is:
A=P(1+(r/n)^nt
A = future value you want to find.
P = $65.00
r = .076 / 12 = .006333333
t = 12 * 12 = 144
the n they are showing is equal to 12.
to compound monthly you take the number of years and multiply by 12 and you take the annual interest rate and divide it by 12.
formula becomes:
A = 65 * (1 + (.076/12))^(12*12)
this becomes:
A = 65 * (1.00633333)^144 = 161.3395912.
In 12 years, george will have that much.
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