Question

For linear programming problem, the objective function is \( z = px + qy \), \( p,q>0 \). If at the corner points \( (0,10) \) and \( (5,5) \), the value of \( z \) are 90 and 60 respectively, then the relation between \( p \) and \( q \) is _____

• A. \( p = 3q \)
• B. \( q = 2p \)
• C. \( q = 3p \)
• D. \( p = 2q \)

Hint:

Plug given corner values into objective function to form equations.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We use the given values at specific coordinates to create a system of linear equations in terms of $p$ and $q$.
Step 2: Formula Application:
1. At (0, 10), $z = 90$: $p(0) + q(10) = 90 \implies 10q = 90 \implies q = 9$. 2. At (5, 5), $z = 60$: $5p + 5q = 60$.
Step 3: Explanation:
Substitute $q = 9$ into the second equation: $5p + 5(9) = 60$ $5p + 45 = 60 \implies 5p = 15 \implies p = 3$. Comparing $p=3$ and $q=9$, we see that $q = 3p$.
Step 4: Final Answer:
The relation is $q = 3p$.