Question #169916

A leading process engineering company bought three (3) major pieces of equipment:


a shell and tube heat exchanger; a storage tank and a three-phase vertical separator.


The company purchased the exact same type of equipment three (3) times within a


ten (10) year period.


Due to audit concerns and missing receipts, you are tasked with determining the cost of EACH

piece of equipment in $US, using The Gauss-Jordan Matrix Method. You are only provided with the total cost(US$) of the following:

1st Time: 3 tanks + 4 heat exchangers + 2 vertical separators cost US$157,500


2nd Time: 2 tanks + 1 heat exchanger + 3 vertical separators cost US$90,000


3rd Time: 5 tanks + 3 heat exchangers + 2 vertical separators cost US$149,500




1
Expert's answer
2021-03-09T08:07:58-0500

Let a tank costs xx $US, a heat exchanger costs yy $US, and a vertical separator costs zz $US. Then


3x+4y+2z=1575002x+y+3z=900005x+3y+2z=149500\begin{matrix} 3x+4y+2z=157500 \\ 2x+y+3z=90000 \\ 5x+3y+2z=149500 \\ \end{matrix}

Then


A=(34215750021390000532149500)A=\begin{pmatrix} 3 & 4 & 2 & & 157500 \\ 2 & 1 & 3 & & 90000 \\ 5 & 3 & 2 & & 149500 \\ \end{pmatrix}

Divide row 1 by 3 (R1=R13)(R_1=\dfrac{R_1}{3})


(14/32/35250021390000532149500)\begin{pmatrix} 1 & 4/3 & 2/3 & & 52500 \\ 2 & 1 & 3 & & 90000 \\ 5 & 3 & 2 & & 149500 \\ \end{pmatrix}

Subtract row 1 multiplied by 2 from row 2 (R2=R2(2)R1)(R_2=R_2-(2)R_1)


(14/32/35250005/35/315000532149500)\begin{pmatrix} 1 & 4/3 & 2/3 & & 52500 \\ 0& -5/3 & 5/3 & & -15000 \\ 5 & 3 & 2 & & 149500 \\ \end{pmatrix}

Subtract row 1 multiplied by 5 from row 3 (R3=R3(5)R1)(R_3=R_3-(5)R_1)


(14/32/35250005/35/315000011/34/3113000)\begin{pmatrix} 1 & 4/3 & 2/3 & & 52500 \\ 0& -5/3 & 5/3 & & -15000 \\ 0 & -11/3& -4/3 & & -113000 \\ \end{pmatrix}

Multiply row 2 by 35-\dfrac{3}{5} (R2=(35)R2)(R_2=(-\dfrac{3}{5})R_2)


(14/32/3525000119000011/34/3113000)\begin{pmatrix} 1 & 4/3 & 2/3 & & 52500 \\ 0 & 1 &-1 & & 9000 \\ 0 & -11/3& -4/3 & & -113000 \\ \end{pmatrix}

Subtract row 2 multiplied by 43\dfrac{4}{3} from row 1 (R1=R1(43)R2)(R_1=R_1-(\dfrac{4}{3})R_2)


(102405000119000011/34/3113000)\begin{pmatrix} 1 & 0 & 2 & &40500 \\ 0 & 1 &-1 & & 9000 \\ 0 & -11/3& -4/3 & & -113000 \\ \end{pmatrix}

Add row 2 multiplied by 113\dfrac{11}{3} to row 3 (R3=R3+(113)R2)(R_3=R_3+(\dfrac{11}{3})R_2)


(10240500011900000580000)\begin{pmatrix} 1 & 0 & 2 & &40500 \\ 0& 1 &-1 & & 9000 \\ 0 & 0 & -5 & & -80000 \\ \end{pmatrix}

Divide row 3 by 5-5 (R3=R35)(R_3=\dfrac{R_3}{-5})


(10240500011900000116000)\begin{pmatrix} 1 & 0 & 2 & &40500 \\ 0& 1 &-1 & & 9000 \\ 0 & 0 & 1 & & 16000 \\ \end{pmatrix}

Subtract row 3 multiplied by 2 from row 1 (R1=R1(2)R3)(R_1=R_1-(2)R_3)


(1008500011900000116000)\begin{pmatrix} 1 & 0 & 0 & &8500 \\ 0& 1 &-1 & & 9000 \\ 0 & 0 & 1 & & 16000 \\ \end{pmatrix}

Add row 3 to row 2 (R2=R2+R3)(R_2=R_2+R_3)


(10085000102500000116000)\begin{pmatrix} 1 & 0 & 0 & &8500 \\ 0& 1 & 0 & & 25000 \\ 0 & 0 & 1 & & 16000 \\ \end{pmatrix}

A tank costs 85008500 $US, a heat exchanger costs 2500025000 $US, and a vertical separator costs 1600016000 $US.



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