Question

When interest is compounded continuously, the amount of money increases at a rate proportional to the amount S present at time t, that is,

dS/dt = rS, where r is the annual rate of interest.
(a) Find the amount of money accrued at the end of 7 years when $6000 is deposited in a savings account drawing 5 and 3/4% annual interest compounded continuously?
(b) In how many years will the initial sum deposited have doubled? (Round your answer to the nearest year.)
(c) Use a calculator to compare the amount obtained in part (a) with the amount S=6000 (1+1/4(0.0575))^7 that is accrued when interest is compounded quarterly.

Answer 1

The amount of money accrued at the end of 7 years when $6000 is deposited in a savings account with continuously compounded interest at a rate of 5 and 3/4% is approximately $8003.94. It will take approximately 12 years for the initial sum deposited to double. The amount accrued when interest is compounded quarterly is approximately $7949.34.

Step-by-step explanation:

To find the amount of money accrued at the end of 7 years when $6000 is deposited in a savings account with continuously compounded interest at a rate of 5 and 3/4%, we can use the formula A = P*e^(rt), where A is the amount of money accrued, P is the principal amount, r is the annual interest rate as a decimal, and t is the time in years.

Using the given values, we have r = 0.0575 and t = 7. Plugging these values into the formula, we get:

A = 6000 * e^(0.0575*7).

Using a calculator to evaluate this expression, we find that the amount of money accrued is approximately $8003.94.

In order to determine how many years it will take for the initial sum deposited to double, we can use the formula A = P*e^(rt), where A is the future amount, P is the principal amount, r is the annual interest rate as a decimal, and t is the time in years. We want to find the value of t when A = 2P. Substituting these values into the formula, we get 2P = P*e^(rt). Dividing both sides by P, we have 2 = e^(rt). Taking the natural logarithm of both sides, we get ln(2) = rt. Solving for t, we get t = ln(2)/r. Using the given interest rate of 5 and 3/4% or 0.0575 as r, we can calculate t as ln(2)/0.0575. Evaluating this expression using a calculator, we find that it will take approximately 12 years for the initial sum to double.

To compare the amount obtained in part (a) with the amount S=6000(1+1/4(0.0575))^7 that is accrued when interest is compounded quarterly, we can substitute the given values into the formula and calculate the difference. Plugging in the values, we have S = 6000(1+1/4(0.0575))^7. Evaluating this expression using a calculator, we find that the amount accrued when interest is compounded quarterly is approximately $7949.34.