The diagonal of a rectangular parallelepiped makes an angle of 30 degrees with one of its lateral

| July 2, 2021 | education

The diagonal of a rectangular parallelepiped makes an angle of 30 degrees with one of its lateral faces, and an angle of 45 degrees with the other. Find the volume if the length of its diagonal is 12 cm.

Consider a triangle DА1В1. Since A1B1 is perpendicular to the plane AA1DD1, then the diagonal A1D is perpendicular to the edge A1D1, and therefore, triangle DA1D1 is rectangular.

The A1B1 leg lies opposite the angle 30, and therefore is equal to half the length of the hypotenuse DB1.

A1B1 = DB1 / 2 = 12/2 = 6 cm.

Since ABCDA1B1C1D1 is a parallelepiped, then A1B1 = D1C1 = AB = DC = 6 cm.

Consider a triangle DC1B1. Since C1B1 is perpendicular to the plane CC1D1D, then the diagonal DC1 is perpendicular to the edge C1D1, and therefore the triangle DC1B1 is rectangular.

Since, by condition, one of the angles in a right-angled triangle is 45, this triangle is isosceles, DC1 = C1B1 = DB1 * Cos45 = 12 * √2 / 2 = 6 * √2 cm.

C1B1 = CB = AD = A1D1 = 6 * √2 cm.

From the right-angled triangle DСС1 we define the leg CC1.

CC1 ^ 2 = DC1 ^ 2 – DC ^ 2 = (6 * √2) ^ 2 – 6 ^ 2 = 72 – 36 = 36.

CC1 = √36 = 6 cm.

Then the volume of the parallelepiped is: V = АD * DC * CC1 = 6 * √2 * 6 * 6 = 216 * √2 cm3.

Answer: The volume of a parallelepiped is 216 * √2 cm3.



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