Which of the cones listed below can be formed from a $252^\circ$ sector of a circle of radius $10$ by aligning the two straight sides? 

$\textbf{(A)} \text{ A cone with slant height of } 10 \text{ and radius } 6$
$\textbf{(B)} \text{ A cone with height of } 10 \text{ and radius } 6$
$\textbf{(C)} \text{ A cone with slant height of } 10 \text{ and radius } 7$
$\textbf{(D)} \text{ A cone with height of } 10 \text{ and radius } 7$
$\textbf{(E)} \text{ A cone with slant height of } 10 \text{ and radius } 8$
The blue lines will be joined together to form a single blue line on the surface of the cone, hence $10$ will be the slant height of the cone.
The red line will form the circumference of the base. We can compute its length and use it to determine the radius.
The length of the red line is $\dfrac{252}{360}\cdot 2\pi \cdot 10 = 14\pi$. This is the circumference of a circle with radius $7$.
Therefore the correct answer is $\boxed{\textbf{(C)} \text{ A cone with slant height of } 10 \text{ and radius } 7}$.