Algebra

Linear Programming

Medium

applied

The corner points of the bounded feasible region for a linear programming problem (LPP) are (0, 3/2), (1, 2) and (4, 0). If the objective function is Z = ax + by, where 'a' and 'b' are positive, then the condition on 'a' and 'b' so that the maximum of Z occurs at (1, 2) and (4, 0) is:

a+2b

2a=b

3a = 2b

3a = 4b

✅ Correct Option: 3

For the maximum of Z to occur at both points (1, 2) and (4, 0) simultaneously, both points must give the same maximum value.

The value of Z at each corner point:

At (0, 3/2): $Z = a(0) + b(\frac{3}{2}) = \frac{3b}{2}$

At (1, 2): $Z = a(1) + b(2) = a + 2b$

At (4, 0): $Z = a(4) + b(0) = 4a$

For maximum to occur at both (1, 2) and (4, 0), these two points must have equal Z values:

$a + 2b = 4a$

$2b = 3a$

$3a = 2b$

When $3a = 2b$ , the value $a + 2b = 4a = \frac{3b}{2} + \frac{b}{2} \geq \frac{3b}{2}$

Since $a, b > 0$ , this confirms the condition for maximum.

Therefore, the condition is $3a = 2b$ .

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Ravi rides his motor cycle at 25 km per hour, he has to spend ₹ 4 per kilometer on petrol, if he rides at a faster speed of 40 km per hour, the petrol cost increases to ₹ 10 per kilometer. He can spend ₹ 200 on petrol and wishes to maximize distance he can travel within one hour. Which of the following is/are TRUE?

A. Maximum distance covered in one hour is 30 km

B. The distance covered at speed of 25km/hr is 50/3 km

C. The distance covered at speed of 25km/hr is $\frac{25}{2}$ km

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Choose the correct answer from the options given below:

The corner points of the bounded feasible region for a linear programming problem (LPP) are (0, 3/2), (1, 2) and (4, 0). If the objective function is Z = ax + by, where 'a' and 'b' are positive, then the condition on 'a' and 'b' so that the maximum of Z occurs at (1, 2) and (4, 0) is:

Solution

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