You deposit $1000 into an account that pays interest at a 5.5% annual rate compounded continuously. After years, the balance in your account is given by the equation . How long will it take for the balance to grow to $2000? Round your answer to the nearest tenth.
1000 e^(.055t) = 2000
e^(.055t) = 2
take the ln of both sides
.055t lne = ln 2, but ln e = 1
solve for t
To find out how long it will take for the balance to grow to $2000, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the ending balance
P = the principal amount (initial deposit)
e = the mathematical constant approximately equal to 2.71828
r = the annual interest rate (in decimal form)
t = the time (in years)
In this case, P = $1000, A = $2000, and r = 5.5% = 0.055 (in decimal form). We need to solve for t.
2000 = 1000 * e^(0.055t)
Dividing both sides of the equation by 1000:
2 = e^(0.055t)
To isolate the exponent and solve for t, take the natural logarithm (ln) of both sides:
ln(2) = ln(e^(0.055t))
Using the property ln(e^x) = x, the equation becomes:
ln(2) = 0.055t
Divide both sides by 0.055 to solve for t:
t = ln(2) / 0.055
Using a calculator, we find that ln(2) is approximately 0.69315. Substituting this value into the equation:
t = 0.69315 / 0.055
t ≈ 12.6
Therefore, it will take approximately 12.6 years for the balance to grow to $2000, rounded to the nearest tenth.
To find out how long it will take for the balance in the account to grow to $2000, we need to solve the equation:
2000 = 1000 * e^(0.055t),
where t represents the time in years.
First, we divide both sides of the equation by 1000 to isolate the exponential term:
2 = e^(0.055t).
Next, we take the natural logarithm of both sides of the equation to remove the exponential term:
ln(2) = ln(e^(0.055t)).
Since ln(e^(0.055t)) is equal to (0.055t), the equation simplifies to:
ln(2) = 0.055t.
Now we can solve for t by dividing both sides of the equation by 0.055:
t = ln(2) / 0.055.
Using a calculator, we can find that ln(2) is approximately 0.693. Therefore, the equation becomes:
t = 0.693 / 0.055.
Evaluating this expression gives us:
t ≈ 12.6.
Rounding to the nearest tenth, it will take approximately 12.6 years for the balance to grow to $2000 in the account.