Find the maximum possible cone volume.

01 / Standard#Your turn

V(x)=\frac\pi3x^2\sqrt{1-x^2},\quad0\le x\le1

Official paper · jm02-2024 · 2(a)(v) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Volume is nonnegative.

Hint 2

Take the positive square root of the maximum of V².

Worked solution

Both endpoint volumes vanish and the interior maximum is larger.
$\max V^2=4\pi^2/243$
Recover volume with cubic-metre units.
$V_{\max}=\sqrt{\frac{4\pi^2}{243}}=\frac{2\sqrt3\pi}{27}\ \mathrm{m}^3$

Maximum volume 2√3π/27 m³.

Checks and common pitfalls: Do not report the squared-volume maximum as a volume.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Both endpoint volumes vanish and the interior maximum is larger.
$\max V^2=4\pi^2/243$
Recover volume with cubic-metre units.
$V_{\max}=\sqrt{\frac{4\pi^2}{243}}=\frac{2\sqrt3\pi}{27}\ \mathrm{m}^3$

Think first. Reveal a hint when the class is ready.

2024 JM02