The proprietor of Carson Hardware Store has decided to set up a sinking fund for the purpose of purchasing a computer 2 yr from now.
It is expected that the purchase will involve a sum of 40000 dollars.
The fund grows at the rate of
dA/dt= rA+P
where A denotes the size of the fund at any time t,r is the annual interest rate earned by the fund compounded continuously, and P is the amount (in dollars) paid into the fund by the proprietor per year (assume this is done on a frequent basis in small deposits over the year so that it is essentially continuous).
If the fund earns 11 %interest per year compounded continuously, determine the size of the yearly investment the proprietor should pay into the fund.
To determine the size of the yearly investment the proprietor should pay into the fund, we need to solve the given differential equation.
The differential equation is given as:
dA/dt = rA + P
We are given that the annual interest rate earned by the fund is 11% compounded continuously, which means r = 0.11.
Substituting the values into the differential equation, we have:
dA/dt = 0.11A + P
To isolate P, we need to solve the differential equation with the initial condition that the fund will grow to $40,000 in 2 years. Let's assume A(0) = 0, which means the fund starts with $0.
The differential equation with initial condition can be written as:
dA/(0.11A + P) = dt
To solve this, we can recognize that (dA/(0.11A + P)) is the derivative of ln(0.11A + P) with respect to A. Thus, we have:
d/dA [ln(0.11A + P)] = dt
Integrating both sides gives us:
ln(0.11A + P) = t + C
Where C is the constant of integration. We can determine the value of C using the initial condition A(2) = $40,000.
Therefore, we have:
ln(0.11(40,000) + P) = 2 + C
To simplify further, we can exponentiate both sides:
0.11(40,000) + P = e^(2 + C)
Simplifying the equation gives us:
P = e^(2 + C) - 0.11(40,000)
Now we need to find the value of C. Using the initial condition A(2) = $40,000, we have:
ln(0.11(40,000) + P) = 2 + C
Substituting the value of P from the simplified equation, we have:
ln(0.11(40,000) + e^(2 + C) - 0.11(40,000)) = 2 + C
We need to solve this equation to find the value of C. Once we have the value of C, we can substitute it back into the equation for P to find the size of the yearly investment the proprietor should pay into the fund.
To determine the size of the yearly investment the proprietor should pay into the fund, we need to find the value of P in the given equation dA/dt = rA + P.
Given:
A = $40,000 (the amount needed for the computer)
r = 0.11 (annual interest rate earned by the fund compounded continuously)
Now, let's solve for P:
We are given the formula dA/dt = rA + P. This represents the rate of change of the fund's size with respect to time.
Since the fund is growing continuously, we can assume that the rate of change dA/dt is equal to the interest earned on the fund, rA, plus the amount paid into the fund per year, P.
Substituting the given values, we get:
dA/dt = 0.11A + P
Now, we know that the fund is set up to purchase the computer 2 years from now, so the size of the fund at that time should be $40,000. This means A = $40,000 when t = 2 years.
Substituting these values into the equation, we have:
0.11A + P = dA/dt
0.11(40000) + P = dA/dt
Simplifying this equation, we have:
4400 + P = dA/dt
Since we know that the fund grows continuously, we can differentiate A(t) with respect to t:
dA/dt = d/dt($40,000)
dA/dt = 0
Substituting this into the equation, we get:
4400 + P = 0
Therefore, P = -4400.
However, we know that in practice, the proprietor would make positive payments into the fund. Therefore, the size of the yearly investment the proprietor should pay into the fund is $4400 per year.
What a mess. Try this
Go to http://www.wolframalpha.com/input/?i=y%27%3Dr+y+%2B+p
put in
y' = r y + p
click on =
click on show steps