You invest $3000 in a savings account where you earn 4% interest compounded continuously
a) what is the formula A(t) for the balance after t years?
A(t)=3000e^0.04(t)
b) How long will it take for the money to double?
6000=3000e^0.04t Is this the correct setup so far or would I just plug in 2 for t?
no, don't sub in t=2
you already took care of the "doubling" when you changed the 3000 to 6000
so you have to solve
6000 = 3000e^(.04t) for t
divide both sides by 3000,
2 = e^(.04t)
take ln of both sides
ln 2 = ln (e^(.04t))
ln 2 = .04t (ln e), but lne = 1
so
.04t = ln 2
t = ln 2/.04 = 17.33 years
dx/dt = .04x
ln x = .04 t + C
x = e^(.04t + C) = ce^(.04 t)
c = 3000
so
x = 3000 e^(.04 t) check
6000/3000 = 2 ON THE LEFT
2 = e^(.04 t)
ln 2 = .04 t
etc
The correct setup for finding how long it will take for the money to double is:
6000 = 3000e^(0.04t)
To solve for t, you need to isolate the variable t. Divide both sides of the equation by 3000:
2 = e^(0.04t)
Now, take the natural logarithm of both sides of the equation:
ln(2) = ln(e^(0.04t))
Using the property ln(e^x) = x, simplify the equation:
ln(2) = 0.04t
Finally, divide both sides of the equation by 0.04 to solve for t:
t = ln(2) / 0.04
You can plug this value into a calculator to find the approximate time it will take for the money to double.
Yes, you have set up the equation correctly. To find the time it takes for the money to double, you set the balance A(t) equal to double the initial amount, which is $6000 in this case. The equation becomes:
6000 = 3000e^(0.04t)
To solve for t, you can begin by dividing both sides of the equation by 3000:
2 = e^(0.04t)
Next, you can take the natural logarithm (ln) of both sides of the equation to isolate the exponential term:
ln(2) = ln(e^(0.04t))
By the log property, the natural logarithm of the exponential term simplifies to:
ln(2) = 0.04t
Finally, you can solve for t by dividing both sides of the equation by 0.04:
t = (ln(2)) / 0.04
Using a calculator, you can find the approximate value of t to be:
t ≈ 17.33
Therefore, it would take approximately 17.33 years for the money to double in the given savings account.