Differentiate between:
Donor region and Recipient region.
Donor region: A donor region is one that exports or supplies resources, labor, or goods to other regions. These regions typically have a surplus of certain resources that can be distributed.
Recipient region: A recipient region is one that receives resources, labor, or goods from other areas. These regions often lack sufficient resources to meet their demands and rely on external supplies.
The slope of the tangent to the curve \( x = \sin\theta \) and \( y = \cos 2\theta \) at \( \theta = \frac{\pi}{6} \) is ___________.
Solve the following L.P.P. by graphical method:
Maximize:
\[ z = 10x + 25y. \] Subject to: \[ 0 \leq x \leq 3, \quad 0 \leq y \leq 3, \quad x + y \leq 5. \]