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EXPLANATION.
Linear programming.
⇒ (max)z = 10500x + 9000y.
⇒ x + y ≤ 50.
⇒ 20x + 10y ≤ 800.
⇒ x ≥ 0, y ≥ 0.
As we know that,
We can write equation as,
⇒ x + y = 50. - - - - - (1).
⇒ 20x + 10y = 800. - - - - - (2).
From equation (1), we get.
⇒ x + y = 50. - - - - - (1).
Put the value of x = 0 in equation, we get.
⇒ (0) + y = 50.
⇒ y = 50.
Their Co-ordinates = (0,50).
Put the value of y = 0 in equation, we get.
⇒ x + (0) = 50.
⇒ x = 50.
Their Co-ordinates = (50,0).
From equation (2), we get.
⇒ 20x + 10y = 800. - - - - - (2).
Put the value of x = 0 in equation, we get.
⇒ 20(0) + 10y = 800.
⇒ 10y = 800.
⇒ y = 80.
Their Co-ordinates = (0,80).
Put the value of y = 0 in equation, we get.
⇒ 20x + 10(0) = 800.
⇒ 20x = 800.
⇒ x = 40.
Their Co-ordinates = (40,0).
Both curves intersects at a point = (30,20).
Put the all points in the equation and check the maximum values, we get.
⇒ (max)z = 10500x + 9000y.
(1) = Co-ordinates = (0,50).
⇒ (max)z = 10500 x 0 + 9000 x 50.
⇒ (max)z = 450000.
(2) = Co-ordinates = (50,0).
⇒ (max)z = 10500 x 50 + 9000 x 0.
⇒ (max)z = 525000.
(3) = Co-ordinates = (0,80).
⇒ (max)z = 10500 x 0 + 9000 x 80.
⇒ (max)z = 720000.
(4) = Co-ordinates = (40,0).
⇒ (max)z = 10500 x 40 + 9000 x 0.
⇒ (max)z = 420000.
(5) = Co-ordinates = (30,20).
⇒ (max)z = 10500 x 30 + 9000 x 20.
⇒ (max)z = 315000 + 180000.
⇒ (max)z = 495000.
The equation : (max)z = 10500x + 9000y at (30,20) = 495000.
