Answer:
Here, the given function,
P = 5x + 8y,
Subject of constraints,
2x + 3y ≥ 15
, 3x + 2y ≥ 15, x ≥ 0, y ≥ 0,
Graphing 2x + 3y ≥ 15:
The related equation of 2x + 3y ≥ 15 is 2x + 3y = 15,
For x = 0,
2(0) + 3y = 15 ⇒ 3y = 15 ⇒ y = 5,
For y = 0,
2x + 3(0) = 15 ⇒ 2x = 15 ⇒ x = 7.5,
Join the points (0, 5) and (7.5, 0),
'≥' represents the solid line,
For (0, 0), 2(0) + 3(0) ≥ 15( False ),
i.e. shaded region will not contain the origin.
Graphing 3x + 2y ≥ 15:
The related equation of 3x + 2y ≥ 15 is 3x + 2y = 15,
For x = 0,
3(0) + 2y = 15 ⇒ 2y = 15 ⇒ y = 7.5,
For y = 0,
3x + 2(0) = 15 ⇒ 3x = 15 ⇒ x = 5,
Join the points (0, 7.5) and (5, 0),
'≥' represents the solid line,
For (0, 0), 3(0) + 2(0) ≥ 15( False ),
i.e. shaded region will not contain the origin.
By graphing,
We found the feasible region,
Having boundary points (3, 3), (0, 7.5) and (7.5,0),
For (3,3), P = 5(3) + 8(3) = 15 + 24 = 39,
For (0, 7.5), P = 5(0) + 8(7.5) = 60,
For (7.5, 0), P = 5(7.5) + 8(0) = 37.5
Hence, min (P) = 37.5