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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;
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.