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.

Question

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.

Answer 1

1000 e^(.055t) = 2000

e^(.055t) = 2
take the ln of both sides
.055t lne = ln 2, but ln e = 1
solve for t

Answer 2

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.

Answer 3

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.