Nanomaterials are characterized by their very small structures, typically ranging from 1 to 100 nanometers.
These materials are known for their distinctive optical, electrical, and mechanical properties that differ significantly from bulk materials.
One common method for creating nanomaterials is the sol-gel process, which involves the hydrolysis and subsequent condensation of metal alkoxides \( M(OR)_x \).
The reaction can be represented as follows: \[ M(OR)_x + x H_2O \rightarrow M(OH)_x + x ROH \]
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: \[ 0 \leq x \leq 3, \quad 0 \leq y \leq 3, \quad x + y \leq 5. \]