= Solution
In the plane $x=0$, the right half of the cross-section has coordinates $y=a(1-\cos\theta)\sin\theta$, $z=a(1-\cos\theta)\cos\theta$, $0\leq\theta\leq\pi$. Reflect across the $z$ axis for the other half. The <cardioid> has a cusp at $(y,z)=(0,0)$ for $\theta=0$, crosses $z=0$ at $(\pm a,0)$ for $\theta=\pi/2$, and reaches $(0,-2a)$ for $\theta=\pi$. The small upper lobes reach $z=a/4$; the widest section is below the equator.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-2-apple-cross-section.png]
{title=Cardioid cross-section and a horizontal slice of the apple-shaped solid}
{description=Original plot in units of $a$. The shaded lower portion has polar angles above $\pi/2$. The thin marked slice is bounded by horizontal planes, rather than by rays from the origin.}
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