A corporation starts to invest part of its revenue continuously at rate of P dollars per year in a fund for future expansion plans. Assume the fund earns money at an annual interest rate, r, compounded continuously. The rate of growth of the amount A in the fund is given by
dA/dt=rA+P.
Find AA as a function of time for P=200 and r=9%.
In your answer round all coefficients and constants to two decimal places as needed.
To find AA as a function of time, we need to solve the differential equation:
dA/dt = rA + P.
First, we can rewrite the equation in a more general form:
dA/dt - rA = P.
This is a first-order linear homogeneous differential equation. To solve it, we can use an integrating factor.
The integrating factor, denoted by μ(t), is given by the exponential of the integral of the coefficient of A, which in this case is -r:
μ(t) = e^(-rt).
Multiplying both sides of the equation by μ(t), we have:
μ(t) * dA/dt - μ(t) * rA = μ(t) * P.
Now, we can recognize the left side as the product rule for differentiation:
(μ(t) * A)' = μ(t) * P.
Integrating both sides with respect to t:
∫ (μ(t) * A)' dt = ∫ μ(t) * P dt.
This simplifies to:
μ(t) * A = ∫ μ(t) * P dt.
Now, let's find the integral of μ(t) * P:
∫ μ(t) * P dt = P * ∫ e^(-rt) dt.
Integrating e^(-rt) with respect to t gives us:
∫ e^(-rt) dt = (-1/r) * e^(-rt) + C,
where C is the constant of integration.
Therefore, the integral of μ(t) * P is:
P * ∫ e^(-rt) dt = -P/r * e^(-rt) + C.
Now, going back to the previous equation:
μ(t) * A = -P/r * e^(-rt) + C.
Dividing both sides by μ(t), we find:
A = (-P/r * e^(-rt) + C) / μ(t).
Substituting the expression for μ(t) (μ(t) = e^(-rt)), we get:
A = (-P/r * e^(-rt) + C) / e^(-rt).
Simplifying further, we have:
A = -P/r * e^(-rt + rt) + C * e^(-rt)
A = -P/r * e^0 + C * e^(-rt)
A = -P/r + C * e^(-rt).
At this point, we need to consider the initial condition of the problem. Let's assume that when t = 0, A = A0.
Substituting t = 0 and A = A0 into the equation, we get:
A0 = -P/r + C * e^(0)
A0 = -P/r + C.
Solving for C, we find:
C = A0 + P/r.
Substituting this value of C back into the equation for A, we have:
A = -P/r + (A0 + P/r) * e^(-rt).
Finally, we can plug in the given values of P = 200, r = 9% (which is 0.09 in decimal form), and round all coefficients and constants to two decimal places:
A = -200/0.09 + (A0 + 200/0.09) * e^(-0.09t)
A = -2222.22 + (A0 + 2222.22) * e^(-0.09t).
Therefore, AA as a function of time is given by:
AA(t) = -2222.22 + (A0 + 2222.22) * e^(-0.09t), rounded to two decimal places.