The maximum value of z = 4x + 3y, if the feasible region for an LPP i…

The correct answer is Option (2) → 112

The shaded region has corners O(0, 0), A(25, 0), B and C(0, 24). B is where the two boundary lines meet.

The line through (0, 24) and (48, 0), by the intercept form:

    \( \displaystyle \frac{x}{48} + \frac{y}{24} = 1 \)

    \(x + 2y = 48\)

The line through \(\left(0, \frac{400}{9}\right)\) and (25, 0):

    \( \displaystyle \frac{x}{25} + \frac{9y}{400} = 1 \)

    \(16x + 9y = 400\)

Putting x = 48 − 2y in the second equation:

    \(16(48 - 2y) + 9y = 400\)
    \(768 - 23y = 400\)
    \(y = 16,\quad x = 16\)

So B = (16, 16). Values of z = 4x + 3y at the corner points:

    \(z(0, 0) = 0\)
    \(z(25, 0) = 100\)
    \(z(16, 16) = 64 + 48 = 112\)
    \(z(0, 24) = 72\)

The maximum value is 112, at B(16, 16).