Find the volume of the body obtained by rotating a right-angled triangle ABC with legs 15 and 20 around

| June 10, 2021 | education

Find the volume of the body obtained by rotating a right-angled triangle ABC with legs 15 and 20 around the perpendicular to the hypotenuse, carried through the apex of the larger acute angle B

In a right-angled triangle ABC, according to the Pythagorean theorem, we determine the length of the hypotenuse AB. AB ^ 2 = AC ^ 2 + BC ^ 2 = 400 + 225 = 625. AB = 25 cm.

The segment AB is the radius of the circle of the truncated cone obtained during rotation.

Savs = BC * AC / 2 = 15 * 20/2 = 150 cm2.

Also Savs = AB * CH / 2.

CH = 2 * Savs / AB = 2 * 150/25 = 12 cm.

In a right-angled triangle BCH, BH ^ 2 = BC ^ 2 – CH ^ 2 = 225 – 144 = 81.

ВН = OС = 9 cm.

BH is the radius of the upper circle, CH is the height of the cone.

Determine the volume of the truncated cone.

V1 = π * CH (AB ^ 2 + OS ^ 2 + AB * OС) / 3 = π * 12 * (625 + 81 + 225) / 3 = π * 3724 cm3.

Let us determine the volume of the SDV cone.

V2 = π * OC ^ 2 * CH / 3 = π * 81 * 12/3 = π * 324 cm3.

Then the volume of the rotation figure is: V = V1 – V2 = π * 3724 – π * 324 = π * 3400 cm3.

Answer: The volume of the rotation figure is π * 3400 cm3.



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