Problem 3: A Volume Problem In triangle ABC, AB = 25, BC = 16, and AC = 39. If ABC is rotated about its shortest side, what is the
volume of the resultant solid?

9514 1404 393

  1200 ≈ 3769.9 cubic units

Step-by-step explanation:

The area of the triangle is given by Heron's formula:

  s = (16 +25 +39)/2 = 40

  A = √(s(s -16)(s -25)(s -39)) = √14400 = 120

Then the height is ...

  A = 1/2·bh

  h = 2A/b = 2(120)/16 = 15

The centroid of the triangle is located 15/3 = 5 units from the axis of rotation, so it will be rotated through a circle with a circumference of ...

  C = 2πr = 10π . . . units

The volume is the product of this distance and the area of the triangle, so is ...

  V = CA = (10π)(120) = 1200π . . . cubic units

The solid's volume is 1200π ≈ 3769.9 cubic units.

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