Дана четырехугольная призма ABCDA’B’C’D’ с параллелограммом ABCD в основании. Известно, что АВ = а, AD = 2а. BAD = 120, A’BA = B’CB = 90, плоскость (AA’C’C) перпендикулярна (BB’D’D). Вычислите объем призмы.
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The volume of a prism is calculated by the formula ( V = B \cdot h ), where ( B ) is the area of the base and ( h ) is the height of the prism. In the case of the given quadrilateral prism with a parallelogram base ABCD, the area of the base ( B ) can be found by splitting the parallelogram into two congruent triangles along one of its diagonals.
Given that ( AB = a ) and ( AD = 2a ), and the angle ( \angle BAD = 120^\circ ), we can calculate the area of one of those triangles using the formula for the area of a triangle with sides ( a ) and ( 2a ) and an included angle of ( 120^\circ ):
[ B_{\triangle BAD} = \frac{1}{2} \cdot AB \cdot AD \cdot \sin(\angle BAD) = \frac{1}{2} \cdot a \cdot 2a \cdot \sin(120^\circ) ]
[ B_{\triangle BAD} = \frac{1}{2} \cdot a \cdot 2a \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}a^2}{2} ]
The area of the parallelogram ( B ) is twice this area:
[ B = 2 \cdot B_{\triangle BAD} = \sqrt{3}a^2 ]
The height ( h ) of the prism is equal to the length of the segment A’B (since ( \angle A’BA = 90^\circ )) or B’C (since ( \angle B’CB = 90^\circ )).
The condition that plane (AA’C’C) is perpendicular to plane (BB’D’D) suggests that the height of the prism is equal to CC’, which is the same as the length of AA’, since these segments are connecting corresponding vertices of the two parallelogram bases.
To find the volume ( V ), we need to know the height ( h ), which is not directly given. However, we can deduce that the height is the same as the length of the edges A’A and B’B since all these edges are perpendicular to the base ABCD.
Thus, ( h = AA’ = BB’ ). The volume ( V ) of the prism is then:
[ V = B \cdot h = \sqrt{3}a^2 \cdot h ]
Without the specific length of ( h ), we cannot calculate a numerical value for the volume. You would need to provide the height or look for additional information in the problem that relates to the height of the prism.