The maximum value of \(Z = 10x + 25y\) subject to \(0 \le x \le 3\), \(0 \le y \le 3\), \(x + y \le 5\), and the point where it occurs, are
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(95\) at \((2, 3)\)
The line \(x + y = 5\) cuts off a corner of the square \(0 \le x \le 3\), \(0 \le y \le 3\). It meets \(x = 3\) at (3, 2) and \(y = 3\) at (2, 3).
The bounded region has corner points (0, 0), (3, 0), (3, 2), (2, 3) and (0, 3). Values of Z:
\(Z(0, 0) = 0\)
\(Z(3, 0) = 30\)
\(Z(3, 2) = 30 + 50 = 80\)
\(Z(2, 3) = 20 + 75 = 95\)
\(Z(0, 3) = 75\)
The maximum is 95, at (2, 3).