We are given a $252^{\circ}$ sector of a circle with a radius of 10.
To form a cone from this sector, we need to align the two straight sides of the sector.
The slant height of the cone will be the radius of the circle, which is 10.
We can use the formula for the slant height of a cone to determine the base radius.
The formula is:
$slant = \sqrt{base^2 + height^2}$
For option A, the base radius is 6 and the slant height is 10.
Plugging these values into the formula, we have:
$10 = \sqrt{6^2 + 10^2}$
Simplifying, we get:
$10 = \sqrt{36 + 100}$
$10 = \sqrt{136}$
This is not true, so option A is not correct.

For option B, the base radius is 6 and the height is 10.
Using the formula for the slant height of a cone, we have:
$slant = \sqrt{6^2 + 10^2}$
$slant = \sqrt{36 + 100}$
$slant = \sqrt{136}$
This is not true, so option B is not correct.

For option C, the base radius is 7 and the slant height is 10.
Using the formula for the slant height of a cone, we have:
$10 = \sqrt{7^2 + height^2}$
Simplifying, we get:
$10 = \sqrt{49 + height^2}$
Squaring both sides of the equation, we have:
$100 = 49 + height^2$
Subtracting 49 from both sides, we find:
$51 = height^2$
Taking the square root of both sides, we have:
$height = \sqrt{51}$

For option D, the base radius is $X$ and the height is 10.
Using the formula for the slant height of a cone, we have:
$slant = \sqrt{X^2 + 10^2}$
This is not enough information to determine the value of $X$.

For option E, the base radius is 8 and the slant height is 10.
Using the formula for the slant height of a cone, we have:
$10 = \sqrt{8^2 + height^2}$
Simplifying, we get:
$10 = \sqrt{64 + height^2}$
This is not true, so option E is not correct.

Based on our calculations, option C is the correct answer.
Therefore, the value of $X$ is 7.
The answer is: 7
Below is a math problem. Provide a detailed, step-by-step solution.

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

[asy]
draw((5.8,8.1)..(-10,0)--(0,0)--(3.1,-9.5)..cycle);
label("10",(-5,0),S);
label("$252^{\circ}$",(0,0),NE);
[/asy]

A. base radius = 6, slant =10

B. base radius = 6, height =10

C. base radius = 7, slant =10

D. base radius = X, height =10

E. base radius = 8, slant = 10
If we know the answer to the above question is C, what is the value of unknown variable X?

### Answer:
