Question

The corner points of the feasible region determined by the system of linear inequalities are (0, 0), (0, 4), (4, 0), (2, 4) and (0, 5). If the maximum value of Z = ax + by where a, b > 0 occurs at both (2, 4) and (4, 0) then

• A. a = 2b
• B. 2a = b
• C. a = b
• D. 3a = b

Answer 1

The Correct Option is A

To solve for the relationship between a and b given that the maximum value of Z = ax + by occurs at both (2, 4) and (4, 0), follow these steps:

1. Calculate the value of Z at point (2, 4):
   Z₁ = a(2) + b(4) = 2a + 4b.
2. Calculate the value of Z at point (4, 0):
   Z₂ = a(4) + b(0) = 4a.
3. Since both points provide the maximum value of Z, set Z₁ equal to Z₂:
   2a + 4b = 4a.
4. Simplify the equation:
   4b = 4a - 2a
5. Simplify further:
   4b = 2a
6. Divide both sides by 2:
   2b = a
7. Rearrange to find the relationship:
   a = 2b

This shows that the correct relationship between a and b is a = 2b.