The diagonal of the base of the rectangular parallelepiped is 20 cm, and makes an angle of 55 degrees with the side of the base, and 40 degrees with the diagonal of the parallelepiped. find the volume.
Since, by condition, the parallelepiped is rectangular, we define the sides of the rectangle at the base through the length of the diagonal and the angle.
Sin 55 = AC / AD.
AC = AD * Sin 55 = 20 * Sin 55.
Cos 55 = CD / AD.
CD = AD * Cos 55 = 20 * Cos 55.
Then the area of the base of the parallelepiped is:
Sax = AC * CD = 20 * Sin 55 * 20 * Cos 55 = 400 * Sin 55 * Cos 55 = 400 * Sin (55 + 55) / 2 = 200 * Sin1100 = 200 * Sin (900 + 200) = 200 * Cos20.
From the right-angled triangle AD1D, we define the leg DD1, which is the height of the parallelepiped.
tg400 = DD1 / AD.
DD1 = h = AD * tg40 = 20 * tg40.
Then the volume of the parallelepiped is: V = Sbn * h = 200 * Cos20 * 20 * tg40 = 4000 * Cos20 * tg40 ≈ 4000 * 0.94 * 0.84 ≈ 3155 cm3.
Answer: V ≈ 3155 cm3.
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