The base of the straight prism ABCDA1B1C1D1 is the parallelogram ABCD with sides 4cm and 8cm, the angle BAD is 60. The diagonal B1D forms an angle with the base plane of 30. Calculate the area of the lateral surface of the prism.
By the cosine theorem, the square of the side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides by the cosine of the angle between them. In triangle BAD we can find the diagonal BD:
BD ^ 2 = 4 ^ 2 + 8 ^ 2-2 * 4 * 8 * cosBAD = 16 + 64-64 * 0.5 = 80-32 = 48;
BD = √48 = 4√3 cm.
By condition, prism ABCDA1B1C1D1 is straight, which means its lateral edges are perpendicular to the bases and equal to the height of the prism.
Consider a triangle B1BD. It is rectangular, B1D is hypotenuse, B1B and BD are legs. Since the diagonal B1D forms an angle of 30 degrees with the base plane, the angle BDB1 = 30 degrees, leg BB1 is opposite to this angle, leg BD is adjacent. The ratio of the opposite leg to the adjacent leg is the tangent of the angle, which means tgBDB1 = BB1 / BD. From here we can find the height of the prism: BB1 = BD * tgBDB1 = BD * tg30 = 4√3 * √3 / 3 = 4 cm.
The area of the lateral surface of a straight prism is equal to the product of the perimeter of its base by the height of the prism:
Sside = P * BB1 = (4 + 4 + 8 + 8) * 4 = 24 * 4 = 96 cm2.
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