The base of a solid in the first quadrant of the xy plane is a right triangle bounded by the coordinate axes and the line x + y = 2. cross sections of the solid perpendicular to the base are squares. what is the volume, in cubic units, of the solid?

Question

Mathematics

Oscar Montgomery

12 August, 06:10

The base of a solid in the first quadrant of the xy plane is a right triangle bounded by the coordinate axes and the line x + y = 2. cross sections of the solid perpendicular to the base are squares. what is the volume, in cubic units, of the solid?

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Jayson Cisneros · Answer 1

The area of an equilateral triangle of side "s" is s^2*sqrt (3) / 4. So the volume of the slices in your problem is

(x - x^2) ^2 * sqrt (3) / 4.

Integrating from x = 0 to x = 1, we have

[ (1/3) x^3 - (1/2) x^4 + (1/5) x^5]*sqrt (3) / 4

= (1/30) * sqrt (3) / 4 = sqrt (3) / 120 = about 0.0144.

Since this seems quite small, it makes sense to ask what the base area might be ... integral from 0 to 1 of (x - x^2) dx = (1/2) - (1/3) = 1/6. Yes, OK, the max height of the triangles occurs where x - x^2 = 1/4, and most of the triangles are quite a bit shorter ...