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Question #41667

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the region in the $xy$ -plane defined by the inequalities $x \ge 0$ , $y \ge 0$ , $x + y \le 4$ , and $2x + y \le 6$ . Let $f(\theta) = \max_{(x,y) \in S} (x \cos \theta + y \sin \theta)$ for $\theta \in \left[0, \frac{\pi}{2}\right]$ . Find the value of the integral $\int_0^{\pi/2} f(\theta) d\theta$ .

Options

A.
$\sqrt{5} + 2\sqrt{2}$
B.
4
C.
$\sqrt{5} + \sqrt{2}$
D.
$2\sqrt{5} + \sqrt{2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Boundary Value Theorem|Linearity and Partitioning|Compound Function Boundaries

Why This Question Is Tricky

Common Mistakes

The problem tests the integration of a compound function derived from a Linear Programming Problem (LPP).

The trap lies in assuming the optimal vertex remains constant across the parameter domain $\theta \in [0, \pi/2]$ .

Students often evaluate the integral using a single 'perceived maximum' vertex (like $(2,2)$ or $(0,4)$ ), ignoring the boundary shifts where the objective function's derivative is discontinuous.

Correctly solving it requires partitioning the integral limits based on the intersection of the vertex projection functions, applying the Boundary Value Theorem implicitly across the piecewise domains.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Boundary Value Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free