Question

The corner points of the feasible region determined by the system of linear inequalities are (0,0), (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 = b
• B. a = 2b
• C. 2a = b
• D. a = 3b

Answer 1

The Correct Option is B

To determine the relationship between \(a\) and \(b\) such that the maximum value of \(Z = ax + by\) occurs at both (2, 4) and (4, 0), we observe that the value of \(Z\) should be equal at these two points.
Let's calculate:
\(Z\) at (2, 4):
\[Z_1 = a \cdot 2 + b \cdot 4 = 2a + 4b\]
\(Z\) at (4, 0):
\[Z_2 = a \cdot 4 + b \cdot 0 = 4a\]
Since \(Z_1 = Z_2\):
\[2a + 4b = 4a\]
Rearranging the equation, we get:
\[4b = 4a - 2a\]
\[4b = 2a\]
Dividing both sides by 2:
\[2b = a\]
Thus, the relationship is \(a = 2b\).