An objective function $z = ax + by$ is maximum at points (15, 15) and (0, 20). If $a, b ≥ 0$ and $ab= 27$, then the maximum value of the objective function is

180

The correct answer is Option (2) → 180

Given: Objective function $z = ax + by$ attains its maximum at two points: $(15, 15)$ and $(0, 20)$

Also given: $a, b \geq 0$ and $ab = 27$

Since the function is maximum at both $(15,15)$ and $(0,20)$, we have:

$z = a(15) + b(15) = 15(a + b)$

$z = a(0) + b(20) = 20b$

Equating both expressions (since both give same maximum value):

$15(a + b) = 20b$

Expanding: $15a + 15b = 20b$ ⟹ $15a = 5b$ ⟹ $a = \frac{b}{3}$

Given: $ab = 27$ ⟹ $\left(\frac{b}{3}\right) \cdot b = 27$

$\frac{b^2}{3} = 27$ ⟹ $b^2 = 81$ ⟹ $b = 9$ (since $b \geq 0$)

Then, $a = \frac{9}{3} = 3$

Now compute maximum value:

$z = 20b = 20 \cdot 9 = 180$