KCET · Maths · Linear Programming
A dietician has to develop a special diet using two foods \(X\) and \(Y\). Each packet (containing \(30 \mathrm{~g}\) ) of food. \(X\) contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food \(Y\) contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240 units of calcium, atleast 460 units of iron and atmost 300 units of cholesterol. The corner points of the feasible region are
- A \((2,72),(40,15),(15,20)\)
- B \((2,72),(15,20),(0,23)\)
- C \((0,23),(40,15),(2,72)\)
- D \((2,72),(40,15),(115,0)\)
Answer & Solution
Correct Answer
(A) \((2,72),(40,15),(15,20)\)
Step-by-step Solution
Detailed explanation
Let \(x\) and \(y\) be the number of packets of food \(X\) and \(Y\)
\[
\begin{array}{ll}
& x \geq 0, y \geq 0 \Rightarrow 12 x+3 y \geq 240 \\
\Rightarrow \quad & 4 x+y \geq 80 \\
& 4 x+20 y \geq 460 \\
\Rightarrow \quad & x+5 y \geq 115 \\
& 6 x+4 y \leq 300 \\
\Rightarrow \quad & 3 x+2 y \leq 150 \\
\Rightarrow \quad & x \geq 0, y \geq 0 \\
& 4 x+y \geq 80
\end{array}
\]
Intersection points of the lines are shown in the graph
we can see that corner point of the feasible region \(A, B\) and \(C\) are \((2,72),(40,15),(15,20)\).
\[
\begin{array}{ll}
& x \geq 0, y \geq 0 \Rightarrow 12 x+3 y \geq 240 \\
\Rightarrow \quad & 4 x+y \geq 80 \\
& 4 x+20 y \geq 460 \\
\Rightarrow \quad & x+5 y \geq 115 \\
& 6 x+4 y \leq 300 \\
\Rightarrow \quad & 3 x+2 y \leq 150 \\
\Rightarrow \quad & x \geq 0, y \geq 0 \\
& 4 x+y \geq 80
\end{array}
\]
Intersection points of the lines are shown in the graph
we can see that corner point of the feasible region \(A, B\) and \(C\) are \((2,72),(40,15),(15,20)\).
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