Answer:
The area of the unpainted region on the four inch board = 160·√3 in.²
Step-by-step explanation:
Here we have that the boards are crossing each other on their flat sides Therefore, when the boards are separated, the area of the unpainted region is the area of a parallelogram
The dimensions of the formed parallelogram are;
Interior angles = 60° and 120° (adjacent angles of a parallelogram)
Height, h of parallelogram formed = 4 inches
From the angle of crossing of the parallelogram, we have;
Angle between the width or perpendicular cross section of the 6 inches board and the angle of crossing of the two boards = 90° - 60° = 30°
Therefore, length of base, b of the parallelogram formed by the unpainted region is given as follows;
![b = \frac{60}{cos(30)} = 40\cdot \sqrt{3} \ inches](https://tex.z-dn.net/?f=b%20%3D%20%5Cfrac%7B60%7D%7Bcos%2830%29%7D%20%20%3D%2040%5Ccdot%20%5Csqrt%7B3%7D%20%5C%20inches)
Therefore, the area of the parallelogram = b × h = 4 × 40·√3 = 160·√3 in.²
Hence, the area of the unpainted region on the four inch board = 160·√3 in.².