 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/12/a/solution

Take the dumbbell axis to be the

z

axis, with its midpoint at

z = 0

. The right-hand sphere has centre

z = α a

R = β a

, while its junction with the cylinder is at

z = γa

. Put

d = (α - γ) a

, so

0 < d < R

. The Pythagorean theorem in a meridional cross-section gives the handle radius

H

from

H^{2} + d^{2} = R^{2}

. Thus

H = a β^{2} - (α - γ)^{2} .

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