Question

A triangle with vertices of (0,0), (0,6) and (5,0) is plotted on the coordinate plane. the

triangle is then rotated about the y-axis to form a solid of revolution. find the volume of
the solid. use t in your calculation and round your final answer to the nearest
hundredth.

Answer 1

50π ≈ 157.08 cubic units

Explanation:

The volume of revolution of a plane figure is the product of the area of the figure and the length of the path of revolution of the centroid of that area. The centroid of a triangle is 1/3 the distance from each side to the opposite vertex. (It is the intersection of medians.)

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length of centroid path

One side of this triangle is the axis of revolution. Then the radius to the centroid is 1/3 the x-dimension of the triangle, so is 5/3. Then the circumference of the circle along which the centroid is revolved is ...

C = 2πr

C = 2π(5/3) = 10π/3 . . . units

triangle area

The area of the triangle is found using the formula ...

A = 1/2bh

A = 1/2(5)(6) = 15 . . . square units

volume

The volume is the product of the area and the path length:

V = AC

V = (15)(10π/3) = 50π . . . cubic units

The volume of the solid is 50π ≈ 157.08 cubic units.

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Additional comment

In the attached figure, the point labeled D is the centroid of the triangle. The label has no significance other than being the next after A, B, C were used to label the vertices.

The volume of revolution can also be found using integration and "shell" or "disc" differential volumes. The result is the same.