 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-1/6/b/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

With

s = sin θ_{0}

and

c = cos θ_{0}

, conversion from spherical coordinates gives

a = r_{0} (s, 0, c)

b = r_{0} (- s /2, 3 s /2, c)

and

c = r_{0} (- s /2, - 3 s /2, c)

V = ∣ det (a, b, c) ∣ = \frac{3 3}{2} r_{0}^{3} sin^{2} θ_{0} ∣ cos θ_{0} ∣.

The absolute value matters when the polar angle is in the lower hemisphere. At the poles or equator the three edge vectors are linearly dependent and the volume vanishes.

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