Chapter 3.5, Problem 58E

a.

To determine

Use a graphing utility to graph each function.

Answer to Problem 58E

  

Explanation of Solution

Calculation:

The amount paid toward the interest is

  $u=M-\left(M-\frac{\mathrm{Pr}}{12}\right){\left(1+\frac{r}{12}\right)}^{12t}$

The amount paid toward the reduction of the principal is

  $v=\left(M-\frac{\mathrm{Pr}}{12}\right){\left(1+\frac{r}{12}\right)}^{12t}$

Where $\text{P}$ the amount of the mortgage is is, $r$ is the interest rate (in decimal form), $\text{M}$ is the monthly payment, and $t$ is the time in years.

The given functions can be simplified as,

  $\begin{array}{l}u=839.06-\left(839.06-\frac{120000*0.075}{12}\right){\left(1+\frac{0.075}{12}\right)}^{12t}\\ u=839.06-\left(89.06\right){\left(1.00625\right)}^{12t}\end{array}$

And,

  $\begin{array}{l}v=\left(839.06-\frac{120000*0.075}{12}\right){\left(1+\frac{0.075}{12}\right)}^{12t}\\ v=\left(89.06\right){\left(1.00625\right)}^{12t}\end{array}$

Now, graph the functions as shown below:

  

Hence, the graph is shown above.

b.

To determine

Find the approximate time.

Answer to Problem 58E

  $t=20.7$ Years.

Explanation of Solution

Calculation: The amount paid toward the interest is

  $u=M-\left(M-\frac{\mathrm{Pr}}{12}\right){\left(1+\frac{r}{12}\right)}^{12t}$

The amount paid toward the reduction of the principal is

  $v=\left(M-\frac{\mathrm{Pr}}{12}\right){\left(1+\frac{r}{12}\right)}^{12t}$

Where $\text{P}$ the amount of the mortgage , $r$ is the interest rate (in decimal form), $\text{M}$ is the monthly payment, and $t$ is the time in years.

Let us consider that the monthly payment is evenly divided between interest and principle then,

  $u=v$

  $839.06-\left(89.06\right){\left(1.00625\right)}^{12t}=\left(89.06\right){\left(1.00625\right)}^{12t}$

On dividing both sides by $89.06$

  $\frac{839.06}{89.06}-{\left(1.00625\right)}^{12t}={\left(1.00625\right)}^{12t}$

Adding ${\left(1.00625\right)}^{12t}$ on both sides

  $\frac{839.06}{89.06}=2{\left(1.00625\right)}^{12t}$

Taking natural logarithm of both sides

  $\ln \frac{839.06}{89.06}=\ln \left(2{\left(1.00625\right)}^{12t}\right)$

  $\ln \frac{839.06}{89.06}=\ln 2+\ln {\left(1.00625\right)}^{12t}$

Subtract $\ln 2$ from both sides,

  $\ln \frac{839.06}{89.06}-\ln 2=12t\ln {\left(1.00625\right)}^{}$

Dividing both sides by $12\ln {\left(1.00625\right)}^{}$

Hence, $t=20.7$ years.

c.

To determine

Conclude for a repayment period of $\text{2}0$ years.

Answer to Problem 58E

As the year decreases , the monthly payment increases, the graph becomes much more linear.

Explanation of Solution

Calculation:

The amount paid toward the interest is

  $u=M-\left(M-\frac{\mathrm{Pr}}{12}\right){\left(1+\frac{r}{12}\right)}^{12t}$

The amount paid toward the reduction of the principal is

  $v=\left(M-\frac{\mathrm{Pr}}{12}\right){\left(1+\frac{r}{12}\right)}^{12t}$

Where $\text{P}$ the amount of the mortgage, $r$ is the interest rate (in decimal form), $\text{M}$ is the monthly payment, and $t$ is the time in years.

The amount and interest paid towards the principle can be written as,

  $\begin{array}{l}u=966.71-\left(839.06-\frac{120000*0.075}{12}\right){\left(1+\frac{0.075}{12}\right)}^{12t}\\ u=966.71-\left(89.06\right){\left(1.00625\right)}^{12t}\end{array}$

  $\begin{array}{l}v=\left(966.71-\frac{120000*0.075}{12}\right){\left(1+\frac{0.075}{12}\right)}^{12t}\\ v=\left(216.71\right){\left(1.00625\right)}^{12t}\end{array}$

Now, graph corresponding to the function is,

  

Let us consider that the monthly payment is evenly divided between interest and principle then,

  $u=v$

  $996.71-\left(216.71\right){\left(1.00625\right)}^{12t}=\left(216.71\right){\left(1.00625\right)}^{12t}$

On dividing both sides by $216.71$

  $\frac{996.71}{216.71}-{\left(1.00625\right)}^{12t}={\left(1.00625\right)}^{12t}$

Adding ${\left(1.00625\right)}^{12t}$ on both sides

  $\frac{996.71}{216.71}=2{\left(1.00625\right)}^{12t}$

Taking natural logarithm of both sides

  $\ln \frac{996.71}{216.71}=\ln \left(2{\left(1.00625\right)}^{12t}\right)$

  $\ln \frac{996.71}{216.71}=\ln 2+\ln {\left(1.00625\right)}^{12t}$

Subtract $\ln 2$ from both sides,

  $\ln \frac{996.71}{216.71}-\ln 2=12t\ln {\left(1.00625\right)}^{}$

Dividing both sides by $12\ln {\left(1.00625\right)}^{}$

  $t=10.7$ years.

Hence, we conclude from both the graphs,we can see that as the year decreases , the monthly payment increases, the graph becomes much more linear.