if tina deposits $500 into a bank account that earns 4.5% interest compounded continuously, how long will it take for her account to have $2500 in it
r = 0.045
Ai = 500
dA/dt = Ar
dA/A = r dt
ln A = r t
A = Ai e^rt
2500 = 500 e^.045 t
5 = e^.045 t
ln 5 = .045 t
so
t = ln 5/.045 =1.61/.045
= 35.8 years
To find out how long it will take for Tina's account to have $2500, we need to use the continuous compounding formula:
A = P * e^(rt)
Where:
A is the final amount in the bank account
P is the principal amount (initial deposit)
e is the mathematical constant approximately equal to 2.71828
r is the interest rate (in decimal form)
t is the time (in years)
We know that Tina deposited $500 into the bank account, so P = $500. The interest rate is 4.5% or 0.045 in decimal form.
Let's plug in the values into the formula:
$2500 = $500 * e^(0.045t)
To solve for t, we need to isolate it. Divide both sides of the equation by $500:
5 = e^(0.045t)
Now, take the natural logarithm (ln) of both sides to eliminate the exponential:
ln(5) = ln(e^(0.045t))
Using the property of logarithms, ln(e^x) = x:
ln(5) = 0.045t
Lastly, divide both sides by 0.045 to solve for t:
t = ln(5) / 0.045
Using a calculator, we can find that ln(5) ≈ 1.609 and divide it by 0.045:
t ≈ 1.609 / 0.045
t ≈ 35.75
Therefore, it will take approximately 35.75 years for Tina's account to have $2500, assuming continuous compounding.