Q:

Find the maximum value of the function for the polygonal convex set determined by the given system of inequalities (Picture provided)

Accepted Solution

A:
Answer:- The maximum value is 86 occurs at (8 , 7)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∵ 3x + 4y = 19 ⇒ (1)∵ -3x + 7y = 25 ⇒ (2)- Add (1) and (2)∴ 11y = 44 ⇒ divide both sides by 11∴ y = 4 ⇒ substitute this value in (1)∴ 3x + 4(4) = 19∴ 3x + 16 = 19 ⇒ subtract 16 from both sides∴ 3x = 3 ⇒ ÷ 3∴ x = 1- One vertex is (1 , 4)∵ 3x + 4y = 19 ⇒ (1)∵ -6x + 3y = -27 ⇒ (2)- Multiply (1) by 2 ∴ 6x + 8y = 38 ⇒ (3)- Add (2) and (3)∴ 11y = 11 ⇒ ÷ 11∴ y = 1 ⇒ substitute this value in (1)∴ 3x + 4(1) = 19 ∴ 3x + 4 = 19 ⇒ subtract 4 from both sides∴ 3x = 15 ⇒ ÷ 3∴ x = 5- Another vertex is (5 , 1)∵ -3x + 7y = 25 ⇒ (1)∵ -6x + 3y = -27 ⇒ (2)- Multiply (1) by -2 ∴ -6x - 14y = -50 ⇒ (3)- Add (2) and (3)∴ -11y = -77 ⇒ ÷ -11∴ y = 7 ⇒ substitute this value in (1)∴ -3x + 7(7) = 25 ∴ -3x + 49 = 25 ⇒ subtract 49 from both sides∴ -3x = -24 ⇒ ÷ -3∴ x = 8- Another vertex is (8 , 7)* Now lets substitute them in f(x , y) to find the maximum value∵ f(x , y) = 2x + 10y∴ f(1 , 4) = 2(1) + 10(4) = 2 + 40 = 42∴ f(5 , 1) = 2(5) + 10(1) = 10 + 10 = 20∴ f(8 , 7) = 2(8) + 10(7) = 16 + 70 = 86- The maximum value is 86 occurs at (8 , 7)