How to calculate the RGB intensity values at center of face using flat shading?
You shouldn't solve for $\cos^{-1}(\theta)$.

The ambient component is as follows (we are assuming the ambient light is white):
$$ambient\ color = (ambient\ light\ color)(ambient\ intensity)(diffuse\ color) = (1,1,1)(0.1)(0.6,0.7,0.8) = (0.06,0.07,0.08)$$

The diffuse component is as follows:
$$diffuse\ color = (light\ color)(light\ intensity)(diffuse\ color)(\cos(\theta)) = (1,1,1)(1)(0.6,0.7,0.8)(0.8) = (0.48,0.56,0.64)$$

Now we just need to add them up:
$$final\ intensity = (diffuse\ color) + (ambient\ color) = (0.06,0.07,0.08) + (0.48,0.56,0.64) = (0.54,0.63,0.72)$$

For more information, here's a pretty decent summary of various lighting calculations.