Francois and Camille Merrit recently inherited a substantial amount of money from a deceased relative. They want to use part of this money to establish an account to pay for their daughter's college education. Their daughter, Lisa, will be starting college five years from now. The Merrits estimate that her first year college expenses will amount to $$14,000$ and increase $$3,000$ per year during each of the remaining three years of her education. The following investments are available to the Merrits.

InvestmentAvailableMaturesReturn of MaturityAEvery year$1$ year$6\%$B$1,3,5,72$ years$14\%$C$1,43$ years$18\%$D$17$ years$65\%$

The Merrits want to determine an investment plan that will provide the necessary funds to cover Lisa's anticipated college expenses with the smallest initial investment. $($Let Ai $=$ investment in year i into investment A in thousands of dollars, Bi $=$ investment in year i into investment B in thousands of dollars, Ci $=$ investment in year i into investment C in thousands of dollars, and Di $=$ investment in year i into investment D in thousands of dollars.$)$

$($a$)$

Formulate an LP model for this problem to find the smallest initial investment $($in thousands of dollars$).$

MIN:

$$A$1+$B$1+$C$1+$D$1$

Your answer includes $2$ characters that can't be graded.

Delete your recent changes and use the pad tools to finish your answer. More information

Subject to:Cash Flow Year $2$

$1\mathrm{.}06$A$11$A$2=0$

Cash Flow Year $3$

$1\mathrm{.}14$B$1+1\mathrm{.}06$A$21$A$31$B$3=0$

Cash Flow Year $4$

$1\mathrm{.}18$C$1+1\mathrm{.}06$A$31$A$41$C$4=0$

Cash Flow Year $5$

$1\mathrm{.}14$B$3+1\mathrm{.}06$A$41$A$51$B$5=0$

Your answer includes $2$ characters that can't be graded.

Delete your recent changes and use the pad tools to finish your answer. More information

Cash Flow Year $6$

$1\mathrm{.}06$A$51$A$6=13$

Cash Flow Year $7$

$1\mathrm{.}18$C$4+1\mathrm{.}14$B$5+1\mathrm{.}06$A$61$A$71$B$7=16$

Your answer includes $1$ character that can't be graded.

Delete your recent changes and use the pad tools to finish your answer. More information

Cash Flow Year $8$

$1\mathrm{.}06$A$5+1\mathrm{.}65$D$11$A$8=19$

Cash Flow Year $9$

Ai$,$ Bi$,$ Ci$,$ Di $0$

$($b$)$

Create a spreadsheet model for this problem and solve it using Solver. What is the optimal solution? $($Round your answers to three decimal places.$)$

$($A$1,$ A$2,$ A$3,$ A$4,$ A$5,$ A$6,$ A$7,$ A$8,$ B$1,$ B$3,$ B$5,$ B$7,$ C$1,$ C$4,$ D$1)=$