ABCA1B1C1 is a straight triangular prism, AC = BC = 5cm, AB = 6cm

| September 11, 2021 | education

ABCA1B1C1 is a straight triangular prism, AC = BC = 5cm, AB = 6cm, HD is perpendicular to AC, BC1D angle is 30 degrees Find the volume of the prism.

Let the length of the segment CD = X cm, then AD = (5 – X) cm.

In right-angled triangles ABD and CBD, we express the leg BD.

BD ^ 2 = AB ^ 2 – AD ^ 2 = 36 – (5 – X) ^ 2.

ВD ^ 2 = ВС ^ 2 – СD ^ 2 = 25 – X ^ 2.

Let’s equate about equality.

36 – (5 – X) ^ 2 = 25 – X ^ 2.

36 – 25 + 10 * X – X ^ 2 = 25 – X ^ 2.

10 * X = 14.

X = CD = 14/10 = 1.4 cm.

Then BD ^ 2 = BC ^ 2 – CD ^ 2 = 25 – 1.96 = 23.04.

BD = 4.8 cm.

In a right-angled triangle BDC1, the leg BD lies opposite the angle 30, then BC1 = 2 * BD = 2 * 4.8 = 9.6 = 48/5 cm.

Then, by the Pythagorean theorem, in a right-angled triangle BCC1:

CC1 ^ 2 = BC1 ^ 2 – BC ^ 2 = (48/5) ^ 2 – 5 ^ 2 = (2304 – 625) / 25 = 1679/25.

CC1 = √1679 / 5 ≈ 8.2 cm.

Sb = BD * AC / 2 = 4.8 * 5/2 = 12 cm2.

Vpr = Sbn * CC1 = 12 * 8.2 = 98.4 cm3.

Answer: The volume of the prism is 98.4 cm3.



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