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I know this question is a little long, but please take the time to help me!

A company makes three blends of fruit drink: Orange Delight (which is 50% orange juice and 50% apple juice); Tropical Sunshine (50% orange juice, 30% mango juice and 20% apple juice); and Mango Mix (60% mango juice, 30% apple juice and 10% orange juice).

(a) Its factories have a total capacity to produce 40000 litres per week.

(b) There is a limited supply of mango juice, but as much apple and orange juice as required. The company has a contract to receive exactly 7500 litres of Mango juice each week. This juice cannot be stored.

(c) The Marketing Department can find buyers for all 40000 litres each week, however optimal demand requires the amount of Orange Delight to be four times as much as the amount of Mango Mix.

Using the variables: d = number of litres of Orange Delight; s = number of litres of Tropical Sunshine; and m = number of litres of Mango Mix, use each of the constraints (a), (b) and (c) to write a linear equation. Form an augmented matrix for this set of linear equations, then find the reduced row echelon form of the matrix and determine how much of each type of fruit drink should be produced each week.

The answer is supposed to be d= 20,000 s = 15,000 m = 5,000.

This is my working but it is way off.

d = number of litres of Orange Delight;

s = number of litres of Tropical Sunshine;

m = number of litres of Mango Mix, use each of the constraints:

d + s + m = 40000

7500 = 0 d + 3 s + 6 m

d = 4s

Standard form: d + m + s = 40000; -3/5m-3/10s = -7500; d-4s = 0;

1, 1, 1, 40000

0, -3/5, -3/10, -7500

1, 0, -4, 0

1, 0, 0, 220000/9

0, -3/5, 0, -17000/3

0, 0, -9/2, -27500

Solution: d = 220000/9; m = 85000/9; s = 55000/9;

(d, m, s) = (24444.4444, 9444.4444, 6111.1111)

Please help me with this question!

Megan Reynolds
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1 Answers1

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You've mixed your juices here: $$d - 4\color{red}{s} = 0.$$ It should have been $$d - 4\color{green}{m} = 0.$$

Denoting the juices $d$, $s$, and $m$ (in that order), here is the system:

\begin{align*} \left[\begin{array}{ccc|r} 1 & 1 & 1 & 40000 \\ 0 & 3/10 & 6/10 & 7500 \\ 1 & 0 & -4 & 0 \end{array}\right] &\rightarrow \left[\begin{array}{ccc|r} 1 & 1 & 1 & 40000 \\ 0 & 1 & 2 & 25000 \\ 1 & 0 & -4 & 0 \end{array}\right] \\ &\rightarrow \left[\begin{array}{ccc|r} 1 & 1 & 1 & 40000 \\ 0 & 1 & 2 & 25000 \\ 0 & 1 & 5 & 40000 \end{array}\right] \\ &\rightarrow \left[\begin{array}{ccc|r} 1 & 1 & 1 & 40000 \\ 0 & 1 & 2 & 25000 \\ 0 & 0 & 3 & 15000 \end{array}\right] \\ &\rightarrow \left[\begin{array}{ccc|r} 1 & 1 & 1 & 40000 \\ 0 & 1 & 2 & 25000 \\ 0 & 0 & 1 & 5000 \end{array}\right] \\ &\rightarrow \left[\begin{array}{ccc|r} 1 & 1 & 0 & 35000 \\ 0 & 1 & 0 & 15000 \\ 0 & 0 & 1 & 5000 \end{array}\right] \\ &\rightarrow \left[\begin{array}{ccc|r} 1 & 0 & 0 & 20000 \\ 0 & 1 & 0 & 15000 \\ 0 & 0 & 1 & 5000 \end{array}\right] \end{align*}

Vedran Šego
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