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Mathematics · 2024 · 4 marks
Linear ProgrammingLinear Programming Problem and its Mathematical Formulation4 marksUnderstandcase study
CBSE 2024 · Region 2 · Set 1 · Q37
The month of September is celebrated as the Rashtriya Poshan Maah across the country. Following a healthy and well-balanced diet is crucial in order…
The month of September is celebrated as the Rashtriya Poshan Maah across the country. Following a healthy and well-balanced diet is crucial in order to supply the body with the proper nutrients it needs. A balanced diet also keeps us mentally fit and promotes improved level of energy.
Figure-$\displaystyle 1$ Figure-$\displaystyle 2$ A dietician wishes to minimize the cost of a diet involving two types of foods, food $\displaystyle \mathrm{X}(\mathrm{x} \mathrm{kg})$ and food $\displaystyle \mathrm{Y}(\mathrm{y} \mathrm{kg})$ which are available at the rate of $\displaystyle ₹ 16 / \mathrm{kg}$ and $\displaystyle ₹ 20 / \mathrm{kg}$ respectively. The feasible region satisfying the constraints is shown in Figure-2. On the basis of the above information, answer the following questions :
(i)
Identify and write all the constraints which determine the given feasible region in Figure-2. Case Study - $\displaystyle 3$
Marking-scheme solution
(i)
Constraints are $\displaystyle \mathrm{x}+2 \mathrm{y} \geq 10$
$$\begin{aligned}
& \mathrm{x}+\mathrm{y} \geq $\displaystyle 6$
& $\displaystyle 3$ \mathrm{x}+\mathrm{y} \geq $\displaystyle 8$
& \mathrm{x} \geq $\displaystyle 0$
& \mathrm{y} \geq $\displaystyle 0$
\end{aligned}
$$(ii)
| Corner points | Value of $\displaystyle \mathrm{Z}=16 \mathrm{x}+20 \mathrm{y}$ |
| $\displaystyle \mathrm{A}(10,0)$ | $\displaystyle 160$ |
| $\displaystyle \mathrm{~B}(2,4)$ | $\displaystyle 112$ |
| $\displaystyle \mathrm{C}(1,5)$ | $\displaystyle 116$ |
| $\displaystyle \mathrm{D}(0,8)$ | $\displaystyle 160$ |
The minimum cost is ₹$\displaystyle 112$
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