you deposit 10,000 dollars in an account that pays 4.25% interest compouned continuously. How long will it take for the account balance to reach $15,000?
i have $15,000=$10,000e^(.0425)(t).
is that right? if not please can i get a reasoning if it's wrong, thanks
<<you deposit 10,000 dollars in an account that pays 4.25% interest compouned continuously. How long will it take for the account balance to reach $15,000?
I have $15,000=$10,000e^(.0425)(t). >>
For "coninuous componding" that would be correct, if t is in years.
1.5 = e^0.425t
ln 1.5 = 0.40547 = 0.0425t
t = 9.54 years
yes
now divide both sides by 10000
1.5 = e^(.0425t)
.0425t = ln 1.5
t = 9.54 years
dy/dt = .0425 y
dy/y = .0425 dt
ln y = .0425 t + c
y = e^(.0425 t +c) = e^c e^.0425 t
= C e^.0425 t
at t = 0, y = 10,000 so C = 10,000
y = 10,000 e^.0425 t
so
YES, you are right
1.5 = e^.0425 t
ln 1.5 = .0425 t
.0425 t = .4055
t = 9.54 years
Well, for once we all agree :)
You probably won't find a financial institution declaring a "continuous compounding" interest rate, or doing it that way. The results would be very nearly the same as daily compounding, but, in that case, a different formula would be used.
I suppose they would want to give you back the same amount at 10 am and 2 pm. They only do calculations at night :)
To determine the time it will take for the account balance to reach $15,000, we need to use the formula for compound interest, which is:
A = P * e^(rt)
Where:
A = the final amount (in this case $15,000)
P = the principal amount (in this case $10,000)
e = the mathematical constant approximately equal to 2.71828
r = the interest rate per compounding period (in this case 4.25%, which needs to be converted to a decimal form, so r = 0.0425)
t = the time in years (what we need to find)
Now, let's substitute the given values into the formula:
$15,000 = $10,000 * e^(0.0425t)
To solve for t, we need to isolate it on one side of the equation. Dividing both sides of the equation by $10,000 gives us:
1.5 = e^(0.0425t)
Now, take the natural logarithm (ln) of both sides of the equation:
ln(1.5) = ln(e^(0.0425t))
According to the properties of logarithms, ln(e^(0.0425t)) is equal to 0.0425t, so the equation becomes:
ln(1.5) = 0.0425t
Finally, to solve for t, divide both sides of the equation by 0.0425:
t = ln(1.5) / 0.0425
Using a calculator, you can evaluate ln(1.5) and then divide it by 0.0425 to find the value of t.