Two Concentrates, Three Targets

A sports-nutrition lab keeps exactly two stock concentrates. One litre of each contributes the amounts below, and carries the price shown:

Concentrate	Protein (g per L)	Fibre (g per L)	Sugar (g per L)	Price (rupees per L)
X	$2$	$1$	$3$	$40$
Y	$1$	$4$	$2$	$55$

Treat each concentrate as a vector in $\mathbb{R}^3$ , one coordinate per nutrient:

$x = \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}, \qquad y = \begin{pmatrix} 1 \\ 4 \\ 2 \end{pmatrix}$

A coach orders a drink that must deliver exactly

$b = \begin{pmatrix} 8 \\ 11 \\ 13 \end{pmatrix}$

grams of protein, fibre and sugar. Only these two concentrates may be used, and the volume of each may be any non-negative real number of litres.

The order is fillable — $b$ genuinely lies in $\operatorname{span}\{x,\ y\}$ — and since $x$ and $y$ are linearly independent, exactly one recipe hits all three targets.

What does that blend cost? Give the exact whole number of rupees.

Two Concentrates, Three Targets

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