Minimize the objective function p = 5x + 8y for the given constraints.
x ≥ 0
y ≥ 0
2x + 3y ≥ 15
3x + 2y ≥ 15

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

5x+10 is the correct one

hope this