Question

Sean invests $10,000 at an annual rate of 5% compounded continuously, according to the

formula A= Pe, where A is the amount, P is the principal, e = 2.718, r is the rate of interest, and
tis time, in years.
Determine, to the nearest dollar, the amount of money Sean will have after 2 years.
Answer: $
Determine how many years, to the nearest year, it will take for Sean's initial investment to
double.
Answer:
years

Answer 1

After 2 years, Sean will have approximately $11,051. It will take about 14 years for Sean's initial investment to double using the formula for continuously compounded interest.

Step-by-step explanation:

The question involves the concept of compound interest compounded continuously. To find out how much Sean will have after 2 years, we use the continuous compounding formula A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount, e is the base of natural logarithms (appx 2.718), r is the annual interest rate (decimal), and t is the time (years).

Given the values, P = $10,000, r = 0.05 (because 5% = 0.05), and t = 2 years, we plug these values into the formula:
A = $10,000 * e(0.05*2) = $10,000 * e0.1.
Using a calculator, we find that e0.1 ≈ 1.105170918. Therefore, Sean will have approximately A = $10,000 * 1.105170918 = $11,051, to the nearest dollar, Sean will have $11,051 after 2 years.

To determine when the investment will double, we set A = 2P and solve for t:
2 * $10,000 = $10,000 * e(0.05t). Dividing both sides by $10,000 we get 2 = e(0.05t). Taking the natural logarithm of both sides gives us ln(2) = 0.05t. Solving for t, we get
t = ln(2) / 0.05.
Using a calculator, t ≈ 13.8629436112, which, to the nearest year, is 14 years.