Suppose that $\$ 18,871$ is invested at an interest rate of $6.3 \%$ per year, compounded continuously.
a) Find the exponential function that describes the amount in the account after time $t$, in years.
b) What is the balance after 1 year? 2 years? 5 years? 10 years?
c) What is the doubling time?
a) The exponential growth function is $P(t)=$
(Type exponential notation with positive exponents. Do not simplify. Use integers or decimals for any numbers in the equation.)
b) The balance after 1 year is $\$$
(Simplify your answers. Round to two decimal places as needed.)
The balance after 2 years is $\$$
(Simplify your answers. Round to two decimal places as needed.)
The balance after 5 years is $\$ \square$.
(Simplify your answers. Round to two decimal places as needed.)
The balance after 10 years is $\$ \square$.
(Simplify your answers. Round to two decimal places as needed.)
c) The doubling time is $\square$ years.
(Simplify your answers. Round to one decimal place as needed.)

Step 1 ：Given that the initial investment (P0) is $18,871 and the annual interest rate (r) is 6.3% or 0.063 in decimal form, we can use the formula for continuous compound interest, which is \(P(t) = P0 * e^{rt}\), where t is the time the money is invested for in years and e is the base of the natural logarithm (approximately equal to 2.71828).

Step 2 ：Substituting the given values into the formula, we get the exponential function that describes the amount in the account after time t as \(P(t) = 18871 * e^{0.063t}\).

Step 3 ：To find the balance after 1 year, we substitute t=1 into the function to get \(P(1) = 18871 * e^{0.063*1}\), which simplifies to \$20098.12 after rounding to two decimal places.

Step 4 ：Similarly, the balance after 2 years is \(P(2) = 18871 * e^{0.063*2}\), which simplifies to \$21405.04 after rounding to two decimal places.

Step 5 ：The balance after 5 years is \(P(5) = 18871 * e^{0.063*5}\), which simplifies to \$25858.16 after rounding to two decimal places.

Step 6 ：The balance after 10 years is \(P(10) = 18871 * e^{0.063*10}\), which simplifies to \$35432.39 after rounding to two decimal places.

Step 7 ：The doubling time is the time it takes for an investment to double in value. It can be found by setting \(P(t) = 2*P0\) and solving for t. Doing so gives us a doubling time of approximately 11.0 years after rounding to one decimal place.