Q. 79 - Precalculus Enhanced with Graphing Utilities

Question

Financial Planning Carletta has \(10,000 to invest. As her financial consultant, you recommend that she invest in Treasury bills that yield 6%, Treasury bonds that yield 7%, and corporate bonds that yield 8%. Carletta wants to have an annual income of \)680, and the amount invested in corporate bonds must be half that invested in Treasury bills. Find the amount in each investment. 

Step-by-Step Solution

Answer

Carletta need to invest $4000 in Treasury bills, $4000 in Treasury bonds and $2000 in corporate bonds.

Step 1. Given information

We have been given that Carletta has $10,000 to invest. As her financial consultant, you recommend that she invest in Treasury bills that yield 6%, Treasury bonds that yield 7%, and corporate bonds that yield 8%. Carletta wants to have an annual income of $680, and the amount invested in corporate bonds must be half that invested in Treasury bills. 

We have to find the amount in each investment. 

Step 2. Assume few variables write the system of equations and corresponding augmented matrix

Let

$$\begin{gathered} \mathrm{x}=\mathrm{Treasury} \mathrm{bills} \\ \mathrm{y}=\mathrm{Treasury} \mathrm{bonds} \\ \mathrm{z}= \mathrm{Corporate} \mathrm{bonds} \end{gathered}$$

$$\begin{gathered} \begin{array}{l}x+y+z\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=10000\\ 0.06x+0.07y+0.08z\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=680\\ z\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\frac{x}{2}\Rightarrow x-2z=0\end{array} \\ Augmented matrix is: \\ \left[\begin{array}{cccc}1& 1& 1& 10000\\ 0.06& 0.07& 0.08& 680\\ 1& 0& -2& 0\end{array}\right] \end{gathered}$$

Step 3. Perform elementary row operations

$$\begin{gathered} R_2\to {\mathrm{R}}_2-0.06{\mathrm{R}}_1: \\ \left[\begin{array}{cccc}1& 1& 1& 10000\\ 0& 0.01& 0.02& 80\\ 1& 0& -2& 0\end{array}\right] \\ {R}_3\to {\mathrm{R}}_3-{\mathrm{R}}_1 \\ \left[\begin{array}{cccc}\mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{10000}\\ \mathrm{0}& \mathrm{0.01}& \mathrm{0.02}& \mathrm{80}\\ \mathrm{0}& \mathrm{-}\mathrm{1}& \mathrm{-}\mathrm{3}& \mathrm{-}\mathrm{10000}\end{array}\right] \\ {R}_{\mathit{2}}\to 0.01{\mathrm{R}}_{\mathit{3}}+\mathrm{R}2: \\ \left[\begin{array}{cccc}\mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{10000}\\ \mathrm{0}& \mathrm{0.01}& \mathrm{0.02}& \mathrm{80}\\ \mathrm{0}& \mathrm{-}\mathrm{1}& \mathrm{-}\mathrm{3}& \mathrm{-}\mathrm{10000}\end{array}\right] \end{gathered}$$

$$\begin{gathered} R_2\leftrightarrow R_3 \\ \left[\begin{array}{cccc}1& 1& 1& 10000\\ 0& -1& -3& -10000\\ 0& 0& -0.01& -20\end{array}\right] \end{gathered}$$

Step 4. Solve for x, y and z

$$\begin{gathered} \begin{array}{r}x+y+z=10000\\ -y-3z=-10000\\ -0.01z=-20\end{array} \\ From last equation, \\ z=\frac{-20}{-0.01} \\ =2000 \end{gathered}$$

Substitute in the second equation.

$$\begin{gathered} -y-3z=-10000 \\ -y-3\cdot 2000=-10000 \\ -y=-10000+6000 \\ -y=-4000 \\ y=4000 \end{gathered}$$

Substitute in the first equation.

$$\begin{gathered} x+y+z=10000 \\ x+4000+2000=10000 \\ x+6000=10000 \\ x=10000-6000 \\ x=4000 \end{gathered}$$

Carletta need to invest $4000 in Treasury bills, $4000 in Treasury bonds and $2000 in corporate bonds.