Question

Question: In 1778, a wealthy Pennsylvanian merchant named Jacob De Haven lent $450,000 to the continental congress to support the troops at valley Forge. The loan was never repaid. Mr De Haven’s descendants have taken the U.S. government to court to collect what they believe they are owed. The going interest rate at the time was 6%. How much were the De Havens owed in 1990

(a) if interest is compounded yearly?

(b) if interest is compound continuously?

Short answer

(a) The interest is $\$104,245,173,00$.

(b) The interest is $ $150,465,982,500$.

Step-by-step solution

(a) Explanation of the solution

An interest is compounded yearly, and to solve a discrete linear dynamical system with one compound as follows.

$$\begin{gathered} x\left(t+1\right)=x\left(t\right)+0.06x\left(t\right) \\ x\left(t+1\right)=1.06x\left(t\right) \end{gathered}$$

The balance after t years is $x\left(t\right)={\left(1.07\right)}^t{x}_0$, where $x_0$ is the amount that Mr De Haven originally lent, that is $\$450,000$.

Then, t is equal to $1990-1778=212$.

Therefore, the De Havens owed in 1990 and as follows.

$$\begin{gathered} x\left(212\right)={\left(1.06\right)}^{212}{x}_0 \\ =231,655.94\left(\$450,000\right) \\ =\$104,245,173,00 \end{gathered}$$

Thus, the compounded yearly interest is $\$104,245,173,00$.

(b) Explanation of the solution

An interest is compounded continuously, and to solve a discrete linear dynamical system with one compound as follows.

$\frac{dx}{dt}=0.06x$

The differential equation separates variables as follows.

$$\begin{gathered} \frac{dx}{dt}=0.06x \\ \frac{dx}{x}=0.06dt \end{gathered}$$

Now, integrating on both sides as follows.

$$\begin{gathered} \int \frac{dx}{x}=\int 0.06dt \\ lnx=0.06t+k \\ x=e^{0.06t+k} \\ x=C{e}^{0.06t} \end{gathered}$$

The initial condition for this equation is 450,000 and so for t=0 as follows.

$$\begin{gathered} x\left(t\right)=C{e}^{0.06t} \\ x\left(0\right)=C{e}^{0.06\left(0\right)} \\ 450000=C \end{gathered}$$

So, the De Havens owed in 1990.

$$\begin{gathered} x\left(t\right)=e^{0.06t}{x}_0 \\ x\left(212\right)=e^{0.06\left(212\right)}\left(\$450000\right) \\ \approx 334,368.85\left(\$450000\right) \\ =150,465,982,500 \end{gathered}$$

Thus, DE Havens owed in 1990 approximately, and the interest compounded continuously is $150,465,982,500$.