Question

An objective function $Z = ax + by$ is maximum at points $(8, 2)$ and $(4, 6)$. If $a \geq 0$ and $b \geq 0$ and $ab = 25$, then the maximum value of the function is:

• A. $60$
• B. $50$
• C. $40$
• D. $80$

Answer 1

The Correct Option is B

The given function \( Z = ax + by \) attains its maximum value at points \((8, 2)\) and \((4, 6)\). At these points:

\[Z_1 = 8a + 2b \quad \text{and} \quad Z_2 = 4a + 6b\]

Since both points yield the same maximum value:

\[8a + 2b = 4a + 6b\]

Simplify the equation:

\[8a - 4a = 6b - 2b\]

\[4a = 4b\]

\[a = b\]

Using the condition \( ab = 25 \):

\[a \cdot b = 25 \quad \text{and} \quad a = b\]

\[a^2 = 25 \implies a = 5 \quad \text{and} \quad b = 5\]

Substitute \( a = 5 \) and \( b = 5 \) into \( Z = ax + by \). At point \((8, 2)\):

\[Z = 8a + 2b = 8(5) + 2(5) = 40 + 10 = 50\]

Thus, the maximum value of \( Z \) is 50.