Solve the problem.
-An initial deposit of $45,000 is made into an account that earns 5.4% compounded continuously. Money is then withdrawn at a constant rate of $2,700 a year until the amount in the account is 0. Find the amount in the
Account at any time t. When is the amount 0? What is the total withdrawn from the account?
A) . The account will be depleted after about 35.1 years. A total of about $94,770 will have been withdrawn at that time.
B) . The account will be depleted after about 42.6 years. A total of about $115,130 will have been withdrawn at that time.
C) . The account will be depleted after about 15.4 years. A total of about $41,580 will have been withdrawn at that time.
D) . The account will be depleted after about 20.5 years. A total of about $55,350 will have been withdrawn at that time.

Question

Solve the problem.
-An initial deposit of $45,000 is made into an account that earns 5.4% compounded continuously. Money is then withdrawn at a constant rate of $2,700 a year until the amount in the account is 0. Find the amount in the
Account at any time t. When is the amount 0? What is the total withdrawn from the account?
A) . The account will be depleted after about 35.1 years. A total of about $94,770 will have been withdrawn at that time. 
B) . The account will be depleted after about 42.6 years. A total of about $115,130 will have been withdrawn at that time. 
C) . The account will be depleted after about 15.4 years. A total of about $41,580 will have been withdrawn at that time. 
D) . The account will be depleted after about 20.5 years. A total of about $55,350 will have been withdrawn at that time.

Quizplus · Accepted Answer

The formula for continuously compounded interest is:
A = Pert
where A is the amount at time t, P is the principal (initial deposit), e is the constant e (approximately 2.71828), r is the annual interest rate as a decimal, and t is the time in years.

Using this formula, we can find the amount in the account at any time t:
A(t) = 45000e^(0.054t)

To find when the account will be depleted, we need to solve for t when A(t) = 0:
0 = 45000e^(0.054t)
Taking the natural logarithm of both sides:
ln(0) = ln(45000) + 0.054t
ln(45000) = -0.054t
t ≈ 42.6 years

To find the total amount withdrawn from the account, we can use the formula for a perpetuity:
P = A/r
where P is the constant annual withdrawal amount, A is the initial amount, and r is the interest rate.

Using this formula, we can find the total amount withdrawn over time:
P = 2700
r = 0.054
A = 45000
Total withdrawn = P(1 - e^(-rt))/r = $115,130 to the nearest dollar.

Therefore, answer B, 11ebb6d7_488d_8552_b437_3986437cf9d2_TB8593_11, is the best choice. The formula for continuous compounding is A(t) = Pe^(rt), where A(t) is the amount at time t, P is the initial deposit, e is Euler's number (approximately 2.71828), r is the interest rate, and t is time in years.

Using this formula, we can find the amount in the account at any time t:
A(t) = 45000*e^(0.054t)
Next, we need to find when the account will be depleted. Let T be the time in years until the account is depleted. We know that the withdrawals will add up to the initial deposit:
2700*T = 45000 - A(T)
Substituting the expression for A(t) into this equation, we get:
2700*T = 45000 - 45000*e^(0.054T)
Dividing both sides by 45000 and simplifying, we get:
e^(0.054T) = 0.8
Taking the natural logarithm of both sides, we get:
0.054T = ln(0.8)
T ≈ 42.6 years

Therefore, the correct answer is B.

Solve the Problem. -An Initial Deposit of $45,000 Is Made into an Account