Question

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?

Answer 1

9514 1404 393

Answer:

1200 ≈ 3769.9 cubic units

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.