Question

The base of a solid in the xy-plane is the circle x2 + y2 = 16. Cross sections of the solid perpendicular to the y-axis are equilateral triangles. What is the volume, in cubic units, of the solid?

Answer 1

c

Explanation:

edge 2021

Answer 2

Recall that the area of an equilateral triangle with side lengths
`s` is
`\frac{\sqrt3}4s^2`. If cross sections are to be taken perpendicular to the y-axis, then each section's side length will be determined by the horizontal distance between the right and left sides of the circle.

Since
`x^2+y^2=16`, you have
`x=\pm√(16-y^2)`, where the positive root corresponds to the right half and the negative roots corresponds to the left half.

The volume is given by


`\displaystyle\int_(-4)^4\frac{\sqrt3}4\left(√(16-y^2)-(-√(16-y^2))\right)^2\,\mathrm dy`

`=\displaystyle\sqrt3\int_(-4)^4(16-y^2)\,\mathrm dy=2\sqrt3\int_0^4(16-y^2)\,\mathrm dy`

where the last equality follows from the fact that the integrand is symmetric about
`y=0`. The volume is then


`2\sqrt3\left(16y-\frac13y^3\right)\bigg|_(y=0)^(y=4)=2\sqrt3\left(64-\frac{4^3}3\right)=(256)/(\sqrt3)`