Question

The maximum value of $z=7x+5y$ subject to $2x+y\le100$, $4x+3y\le240$, $x\ge0$ $y\ge0$ is

• A. 350
• B. 380
• C. 400
• D. 410
• E. 420

Hint:

Linear Programming Tip: The maximum value almost always occurs at the intersection of the two constraint lines rather than the axis intercepts. Testing the intersection point first is a smart move!

Answer 1

The Correct Option is D

Concept:
In Linear Programming, the maximum or minimum value of an objective function ($z$) always occurs at one of the corner points (vertices) of the feasible region defined by the system of inequalities. We must find all intersecting corner points and evaluate $z$ at each one.

Step 1: Find the intersection of the two main constraints.
Treat the inequalities as equations to find where they cross: Equation 1: $2x + y = 100 \implies y = 100 - 2x$ Equation 2: $4x + 3y = 240$ Substitute Equation 1 into Equation 2: $$4x + 3(100 - 2x) = 240$$ $$4x + 300 - 6x = 240$$ $$-2x = -60 \implies x = 30$$ Substitute $x = 30$ back to find $y$: $y = 100 - 2(30) = 40$. Corner Point 1 is $(30, 40)$.

Step 2: Find the x and y intercepts for the feasible region.
For $2x + y = 100$: Intercepts are $(50, 0)$ and $(0, 100)$. For $4x + 3y = 240$: Intercepts are $(60, 0)$ and $(0, 80)$. Because of the $\le$ constraints, the feasible region is bounded by the smallest intercepts on each axis. Valid x-intercept: $(50, 0)$ Valid y-intercept: $(0, 80)$ The origin $(0, 0)$ is also a corner point.

Step 3: List all valid corner points.
The vertices of our bounded feasible region are: 1. $(0, 0)$ 2. $(50, 0)$ 3. $(0, 80)$ 4. $(30, 40)$

Step 4: Evaluate the objective function at each corner point.
Substitute each point into $z = 7x + 5y$: At $(0, 0)$: $z = 7(0) + 5(0) = 0$ At $(50, 0)$: $z = 7(50) + 5(0) = 350$ At $(0, 80)$: $z = 7(0) + 5(80) = 400$ At $(30, 40)$: $z = 7(30) + 5(40) = 210 + 200 = 410$

Step 5: Identify the maximum value.
Compare the calculated $z$ values: $0, 350, 400, 410$. The highest value is 410, which occurs at the point $(30, 40)$. Hence the correct answer is (D) 410.