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Question 1136738: Lia opened a savings account and deposited $500.00. The account earns 8% interest, compounded continuously. If she wants to use the money to buy a new bicycle in 1 year, how much will she be able to spend on the bike?
How would you solve it step by step?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the continuous compounding formula is f = p * e ^ (r * n)
f is the future value
p is the present value
e is the scientific constant of 2.71828828..... usually shown as e^x on your
calculator.
it is the inverse of the natural log function.
by definition y = ln(x) if and only if x = e^y.
ln stands for log to the base e.
for example, if ln(500) = 6.214608098, then e^6.214608098 = 500.
you can use your scientific calculator to confirm.
on my calculator, which is the TI-84 Plus, e^x is 2nd ln.
r is the interest rate per time period (percent / 100 is rate).
n is the number of time periods.
in your problem, the formula of f = p * e^(r * n) becomes:
f = 500 * e ^ (.08 * 1)
p = 500
r = .08 per year
n = 1 year.
simplify this formula to get f = 500 * e ^ (.08).
you should be able to apply this formula directly in your calculator.
in my calculator, i get f = 541.6435338.
in my calculator, i entered 500 * e^(.08).
i actually entered it as 500 * 2nd ln(.08).
when i hit enter, the calculator told me 541.6435338.
if you don't have the e function on your calculator, then you would need to use the constant directly.
for example:
500 * 2.718281828 ^ .08 = 541.6435338.
it's the same answer, or something very close because all the calculator does is translate the e to 2.718182828....
fyi, continuous compounding formula can be simulated by making the number of discrete compounding periods per year very large.
for example, if you made the number of compounding periods equal to 10,00 times a year, then you could use the discrete compounding formula of f = p * (1 + r) ^ n and get an answer that's close.
for example:
f = 500 * (1 + .08/10000) ^ 1000 = 541.6433605 which is pretty close to 541.6435338.
the more discrete compounding periods per year, the close you get to continuous compounding.
note that, with discrete compounding formula, the interest rate per year is divided by the compounding periods per year and the number of year is multiplied by the number of compounding periods per year.
that's why the formuls had the interest rate per time period equal to .08 / 10000 and the number of time periods equal to 1 year * 10,000.
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