= Solution
The <probability density> is symmetric about $300\,\mathrm{mL}$, vanishes at $270$ and $330\,\mathrm{mL}$, and has its single maximum $\pi/(120\,\mathrm{mL})$ at $300\,\mathrm{mL}$. It is zero outside that interval. Its area is one: with the dimensionless <change of variables>
$$
X=\frac{\pi(v-300\,\mathrm{mL})}{60\,\mathrm{mL}},
\qquad p_X(x)=\frac12\cos x,\quad -\frac\pi2\leq x\leq\frac\pi2,
$$
normalization is $\tfrac12\int_{-\pi/2}^{\pi/2}\cos x\,dx=1$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2-q14-coffee.png]
{title=Coffee-volume density and the two-cup tail region}
{description=Original graphs. Left: the density with the passing interval shaded. Right: transformed independent cup volumes $X,Y$ and the triangular region $X+Y\geq\pi/2$ used for the <probability> in part (b).}
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