Chapter 7.3, Problem 93E

Solution

To calculate: The amount invested by him at each rate if he puts at least amount possible in the growth fund.

The amounts for CDs, bonds, and growth funds are $0, $\$38983.05,\$11016.95$ .

Given Information:

Nicolas has $50,000 to invest and wants to receive $5000 interest the first year. He puts the part in CDs earning 5.75% APY, the part in bonds earning 8.7% APY, and the rest in a growth fund earning 14.6% APY.

Calculation:

Consider the given information,

Suppose that x is representing amount invested at 6% APY, and y is representing amount invested at 8% APY, and z is representing amount invested at 10% APY.

Suppose that x is representing amount invested at $5.75\%$ , and y is representing amount invested at $8.7\%$ , and z is representing amount invested at 10% .

Now, write the equation, as Nicolas invested $50000.

  $x+y+z=50000$   ...... (1)

And, he wants a total interest of $5000 in the first years.

  $0.0575x+0.087y+0.146z=5000$   ...... (2)

It can be observed that he wants 10% of the amount invested and the interest rate in CDs is only $5.75\%$ . This indicate to not invest in CDs at all so that only need to divide the money in the two accounts. So,

  $x=0$

Now, write the system of the matrix as $AX=B$ .

  $A=\left[\begin{array}{ccc}1& 1& 1\\ 0.0575& 0.087& 0.146\\ 1& 0& 0\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],\text{ and }B=\left[\begin{array}{c}50000\\ 5000\\ 0\end{array}\right]$

The equation can be solved as,

  $\begin{array}{c}{A}^{-1}AX=A^{-1}B\\ X=A^{-1}B\end{array}$

Solve for  X by multiplying both sides by  $A^{-1}$ . That is,  $X=A^{-1}B$. Using a graphing calculator (as mention in the book), find the solution of the system as shown below:

  

The obtained solutions are defined as,

  $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}0\\ 38983.05\\ 11016.95\end{array}\right]$

Hence, the amounts for CDs, bonds, and growth funds are $0, $\$38983.05,\$11016.95$ .