Answer:
a) The exponential function that describes the amount in the account after time t, in years, when compounded continuously is given by the formula A(t) = P * e^(rt), where:
A(t) = amount after time t
P = initial principal amount ($18,979)
e = Euler's number (approximately 2.71828)
r = annual interest rate (5.1% or 0.051 as a decimal)
t = time in years
So, the exponential function is A(t) = 18979 * e^(0.051t).
b) To find the balance after 1 year, 2 years, 5 years, and 10 years, we can use the exponential function A(t) = 18979 * e^(0.051t) and substitute the respective values of t.
After 1 year: A(1) = 18979 * e^(0.051*1)
After 2 years: A(2) = 18979 * e^(0.051*2)
After 5 years: A(5) = 18979 * e^(0.051*5)
After 10 years: A(10) = 18979 * e^(0.051*10)
c) The doubling time can be found using the formula for exponential growth:
A(t) = P * e^(rt), where A(t) is the amount after time t, P is the initial principal amount, e is Euler's number, r is the annual interest rate, and t is time in years.
To find the doubling time, we can set up the equation:
2P = P * e^(rt), where 2P is the doubled amount.
Solving for t, we get:
2 = e^(rt)
Taking the natural logarithm of both sides gives:
ln(2) = rt
Finally, solving for t gives:
t = ln(2) / r
Substituting the given values, we can find the doubling time:
t = ln(2) / 0.051 ≈ 13.56 years
Therefore, the doubling time is approximately 13.56 years.
Step-by-step explanation: