Question

Question 1 Find the maximum value of the function for the polygonal convex set determined by the given system of inequalities

Answer 1

Answer:

The maximum value is 126 occurs at (9 , 9)

Step-by-step explanation:

* Lets remember that a function with 2 variables can written

 f(x , y) = ax + by + c

- We can find a maximum or minimum value that a function has for

 the points in the polygonal convex set

- Solve the inequalities to find the vertex of the polygon

- Use f(x , y) = ax + by + c to find the maximum value

∵ 8x + 2y = 36 ⇒ (1)

∵ -3x + 6y = 27 ⇒ (2)

- Multiply (1) by -3

∴ -24x - 6y = -108 ⇒ (3)

- Add (2) and (3)

∴ -27x = -81 ⇒ divide both sides by -27

∴ x = 3 ⇒ substitute this value in (1)

∴ 8(3) + 2y = 36

∴ 24 + 2y = 36 ⇒ subtract 24 from both sides

∴ 2y = 12 ⇒ ÷ 2

∴ y = 6

- One vertex is (3 , 6)

∵ 8x + 2y = 36 ⇒ (1)

∵ -7x + 5y = -18 ⇒ (2)

- Multiply (1) by 5 and (2) by -2

∴ 40x + 10y = 180 ⇒ (3)

∴ 14x - 10y = 36 ⇒ (4)

- Add (3) and (4)

∴ 54x = 216 ⇒ ÷ 54

∴ x = 4 ⇒ substitute this value in (1)

∴ 8(4) + 2y = 36

∴ 32 + 2y = 36 ⇒ subtract 32 from both sides

∴ 2y = 4 ⇒ ÷ 2

∴ y = 2

- Another vertex is (4 , 2)

∵ -3x + 6y = 27 ⇒ (1)

∵ -7x + 5y = -18 ⇒ (2)

- Multiply (1) by 7 and (2) by -3

∴ -21x + 42y = 189 ⇒ (3)

∴ 21x - 15y = 54 ⇒ (4)

- Add (3) and (4)

∴ 27y = 243 ⇒ ÷ 27

∴ y = 9 ⇒ substitute this value in (1)

∴ -3x + 6(9) = 27

∴ -3x + 54 = 27 ⇒ subtract 54 from both sides

∴ -3x = -27 ⇒ ÷ -3

∴ x = 9

- Another vertex is (9 , 9)

* Now lets substitute them in f(x , y) to find the maximum value

∵ f(x , y) = 9x + 5y

∴ f(3 , 6) = 9(3) + 5(6) = 27 + 30 = 57

∴ f(4 , 2) = 9(4) + 5(2) = 36 + 10 = 46

∴ f(1 , 5) = 9(9) + 5(9) = 81 + 45 = 126

- The maximum value is 126 occurs at (9 , 9)