In the truncated cone, the radius of the larger base is 12cm, the generatrix is 39cm, the diagonal of the axial section

| January 7, 2021 | education

In the truncated cone, the radius of the larger base is 12cm, the generatrix is 39cm, the diagonal of the axial section is 45cm. Calculate the radius of the smaller base

Let the radius of the circle of the smaller base be equal to X cm.Then BO1 = X cm, OH = OB1 = X cm, and the length of the segment AH = AO – OH = (12 – X) cm, and the length of the segment DН = (12 + X) cm.
From two right-angled triangles, AВН and DВН, we express by the Pythagorean theorem the general leg ВН.
BH ^ 2 = AB ^ 2 – AH ^ 2 = 39 ^ 2 – (12 – X) ^ 2 = 1521 – 144 + 24 * X – X ^ 2 = 1377 + 24 * X – X ^ 2.
ВН ^ 2 = ВD ^ 2 – DН ^ 2 = 45 ^ 2 – (12 + X) ^ 2 = 2025 – 144 – 24 * X – X ^ 2 = 1881 – 24 * X – X ^ 2.
Let’s equate both equalities.
1377 + 24 * X – X ^ 2 = 1881 – 24 * X – X ^ 2.
48 * X = 504.
X = OH = O1B = 504/48 = 10.5 cm.
Answer: The radius of the smaller base is 10.5 cm.



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