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. $80$

Hint:

When solving problems with linear functions like \( Z = ax + by \), it’s crucial to use the given points and conditions to establish relationships between the variables. Simplify the system step by step and use substitution or elimination methods when necessary. Also, remember that when both points yield the same maximum value, you can equate the corresponding expressions and solve for unknowns. This helps find the required values of \( a \) and \( b \), leading to the solution of the problem.

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.

Answer 2

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

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

Step 1: Set the two expressions equal since both yield the same maximum value:

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

Step 2: Simplify the equation:

Subtract \( 4a \) and \( 2b \) from both sides:

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

Step 3: Further simplification:

\[ 4a = 4b \]

Step 4: Solve for \( a \) and \( b \):

\[ a = b \]

Step 5: Use the condition \( ab = 25 \):

We are given that \( ab = 25 \), and from the previous step, we have \( a = b \). Thus:

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

Substitute \( a \) for \( b \) in the equation:

\[ a^2 = 25 \]

Step 6: Solve for \( a \):

\[ a = 5 \quad \text{and} \quad b = 5 \]

Step 7: Substitute \( a = 5 \) and \( b = 5 \) into \( Z = ax + by \) to find the maximum value:

At point \((8, 2)\), substitute the values of \( a \) and \( b \):

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

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