A right cone has slant height 1 m and base radius x m. Derive V²(x) and its radius domain.

01 / Standard#Your turn

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Relate height and radius by Pythagoras.

Hint 2

Square the cone volume formula.

Worked solution

The height is the nonnegative square root.
$h=\sqrt{1-x^2},\quad0\le x\le1$
Substitute into the volume and square.
$V=\frac\pi3x^2h\implies V^2=\frac{\pi^2}{9}(x^4-x^6)$

V²=π²(x⁴−x⁶)/9, 0≤x≤1.

Checks and common pitfalls: The endpoints describe degenerate cones and are included for the closed model.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The height is the nonnegative square root.
$h=\sqrt{1-x^2},\quad0\le x\le1$
Substitute into the volume and square.
$V=\frac\pi3x^2h\implies V^2=\frac{\pi^2}{9}(x^4-x^6)$

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

2024 JM02