Solve the following L.P.P. by graphical method:
Maximize:
\[ z = 10x + 25y. \] Subject to: \[ 0 \leq x \leq 3, \quad 0 \leq y \leq 3, \quad x + y \leq 5. \]
To determine the maximum value of \(Z\), we evaluate it at the vertices of the feasible region.
Objective function:
\[ Z = 10x + 25y \]Evaluating \(Z\) at the given points:
\[ Z \text{ at } O(0,0) = 10(0) + 25(0) = 0 \] \[ Z \text{ at } A(3,0) = 10(3) + 25(0) = 30 \] \[ Z \text{ at } B(3,2) = 10(3) + 25(2) = 30 + 50 = 80 \] \[ Z \text{ at } C(2,3) = 10(2) + 25(3) = 20 + 75 = 95 \] \[ Z \text{ at } D(0,3) = 10(0) + 25(3) = 75 \] Conclusion:The maximum value of \(Z\) is 95 at \(C(2,3)\).
Thus, \(Z\) is maximum when \(x = 2\) and \(y = 3\).
Solve the following LPP graphically: Maximize: \[ Z = 2x + 3y \] Subject to: \[ \begin{aligned} x + 4y &\leq 8 \quad \text{(1)} \\ 2x + 3y &\leq 12 \quad \text{(2)} \\ 3x + y &\leq 9 \quad \text{(3)} \\ x &\geq 0,\quad y \geq 0 \quad \text{(non-negativity constraints)} \end{aligned} \]
Explain the construction of a spherical wavefront by using Huygens' principle.
The slope of the tangent to the curve \( x = \sin\theta \) and \( y = \cos 2\theta \) at \( \theta = \frac{\pi}{6} \) is ___________.