A Microbiologist Must Prepare a Culture Medium in Which to Grow

A microbiologist must prepare a culture medium in which to grow a certain type of bacteria. The percent of salt contained in this medium is given by $S = 12 x y z$ where $x , y,$ and $z$ are the nutrient solutions to be mixed in the medium. For the bacteria to grow, the medium must be 13% salt. Nutrient solutions $x , y,$ and $z$ cost $1, $2, and $3 per liter, respectively. How much of each nutrient solution should be used to minimize the cost of the culture medium?

A) $\begin{array} { l } x = \sqrt [ 3 ] { 0.065 } \approx 0.402 L \\y = \sqrt [ 3 ] { 0.065 } \approx 0.201 L \\z = \sqrt [ 3 ] { 0.065 } \approx 0.134 L\end{array}$
B) $\begin{array} { l } x = \sqrt [ 3 ] { 0.035 } \approx 0.327 L \\y = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \\z = \sqrt [ 3 ] { 0.015 } \approx 0.2466 L\end{array}$
C) $\begin{array} { l } x = \sqrt [ 3 ] { 0.065 } \approx 0.402 L \\y = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \\z = \sqrt [ 3 ] { 0.055 } \approx 0.380 L\end{array}$
D) $\begin{array} { l } x = \sqrt [ 3 ] { 0.025 } \approx 0.292 L \\y = \sqrt [ 3 ] { 0.035 } \approx 0.327 L \\z = \sqrt [ 3 ] { 0.115 } \approx 0.486 L\end{array}$
E) $\begin{array} { l } x = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \\y = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \\z = \sqrt [ 3 ] { 0.265 } \approx 0.642 L\end{array}$