3) Consider the following problem: A hemispherical bubble is placed on a spheric

Question

3) Consider the following problem: A hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical bubble is then placed on the rst one. This process is continued until n chambers, including the sphere, are formed.

a) If there are only 2 bubbles nd the radii of both bubbles. Then nd the maximum height.

b) Can you nd a relationship between each of the radii? (Suppose that the base bubble has a radius of R, nd the next bubbles radius in terms of R.)

c) If there are only 3 bubbles, use part (b) to nd each of the radii. Then nd the maximum height of this bubble tower.

d) Use mathematical induction to prove that the maximum height of any bubble tower with n chambers is given by 1 +n.

Explanation / Answer

Suppose RR is the radius of the nn'th bubble and xx is the radius of the (n+1)(n+1)'th bubble. And suppose dd is the height added when the (n+1)(n+1)'th bubble is added to the top of the tower.

Rn2Rn+12+Rn+1d=RnRn2Rn+12+Rn+1d=Rn

Let d(x)d(x) take Rn+1Rn+1 as the input, and outputs the added height to the tower, dd.

d(x)=R2x2+xRd(x)=R2x2+xR

d(x)=xR2x2+1=0d(x)=xR2x2+1=0

R2x2=xR2x2=x

R2x2=x2R2x2=x2

R2=2x2R2=2x2

22R=x22R=x

22Rn=Rn+1

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