To find the length of the diagonal \( x \) on the bottom face of a cube with a side length of 14, we can use the Pythagorean theorem. In a square, the diagonal can be calculated as follows:
\[ d = \sqrt{s^2 + s^2} \]
where \( s \) is the side length of the square. For this cube:
\[ s = 14 \]
Plugging in the value:
\[ d = \sqrt{14^2 + 14^2} = \sqrt{196 + 196} = \sqrt{392} \]
Now, calculating \( \sqrt{392} \):
\[ \sqrt{392} = \sqrt{4 \times 98} = \sqrt{4} \times \sqrt{98} = 2 \times \sqrt{98} \] \[ \sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2} \]
Thus:
\[ d = 2 \times 7\sqrt{2} = 14\sqrt{2} \]
Next, we find the approximate value of \( \sqrt{2} \):
\[ \sqrt{2} \approx 1.414 \]
Now, substituting back to find \( d \):
\[ d \approx 14 \times 1.414 \approx 19.796 \]
Rounding 19.796 to the nearest tenth gives us:
\[ d \approx 19.8 \]
Therefore, the length of the diagonal \( x \) on the bottom face of the cube is approximately 19.8.