The diagonal of a rectangular parallelepiped is 18 cm and makes an angle of 30 degrees with the plane of the side face and an angle of 45 degrees with the side edge. Find the volume of the parallelepiped.
The angle between the side face and the diagonal is the AC1D linear angle between the DC1 projection and the AC1 diagonal itself.
In a right-angled triangle AC1D, the BP leg lies opposite an angle of 30, then BP = AC1 / 2 = 18/2 = 9 cm. DS1 = AC1 * Cos30 = 18 * √3 / 2 = 9 * √3 cm.
The right-angled triangle ACC1 is isosceles, since the Angle AC1C = 45, then AC = CC1 = AC1 * Sin45 = 18 * √2 / 2 = 9 * √2 cm.
In a right-angled triangle ADC, according to the Pythagorean theorem, CD ^ 2 = AC ^ 2 – AD ^ 2 = 162 – 81 = 81. SD = 9 cm.
Let’s define the volume of the parallelepiped.
V = AD * CC1 * CD = 9 * 9 * √2 * 9 = 729 * √2 cm3.
Answer: The volume of a parallelepiped is 729 * √2 cm3.
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