Unformatted text preview: 432 23. 24. CHAPTER 4 PROBLEMS PLUS dV dr dV dV
_ 4 3 7 2
V — §7rr <:> E — 47W d—t' B.ut E is proportional to the surface area so E_ — k 471'7’2 for some
2 dr 2 dr . . . . . .
constant k. Therefore 47W d—t — —k ~ 4777“ {i} a = k = constant. An ant1der1vat1ve of k With respect to t 18 kt, so 7" 2 [ct + C. Whent : 0. the radius 7" must equal the original radius r0. so C = To. and r 2 kt + 7‘0. To ﬁnd kwe use the fact that whent = 3. 7‘ = 3k +r0 and V 2%1/0 => CAIN; «(3k + To)3 = %  3an => (3k+7°0)3:%7"3 => 3k+ro= _1_,,.0 => k=§r0<L— ).Sincer=kt+r0. €/§ $75
1" : éTo (% — 1) t + To. When the snowball has melted completely we have 7' — 0 :>
3 3
in) (— — 1)t + To — 0 which gives t # 33ﬂ . Hence. it takes 33\/§ — 3 = 3 3 w 11 h 33 min
x/E ﬂ , 1 ﬂ — 1 1/5 — 1
longer. By ignoring the bottom hemisphere of the initial spherical bubble. we can rephrase the problem as follows: Prove
that the maximum height of a stack of n hemispherical bubbles is ﬂ if the radius of the bottom hemisphere is 1.
We proceed by induction. The case n : 1 is obvious since \/T is the height of the ﬁrst hemisphere Suppose the
assertion is true for n : k and let’s suppose we have k + 1 hemispherical bubbles forming a stack of maximum height. Suppose the second hemisphere (counting from the bottom) has radius 7'. Then by our induction hypothesis (scaled to the setting of a bottom hemisphere of radius r), the height of the stack formed by the top Is bubbles is \/E 7'. (If it were shorter, then the total stack of k: + 1 bubbles wouldn’t have maximum height.) The height of the whole stack is H(r) : x/Er + V 1 — r2. (See the ﬁgure) Q We want to choose 1" so as to maximize H (7’) Note that 0 < r < 1. We calculate »—> 4. Ho) 2 f , _1:—r2 and H"(r) : Fiﬁ H'm : 0 e r2=k(1ir2) 4:) (k+1)r2:k 4:) r:‘lk—%.Thisistheonly critical number in (07 1) and it represents a local maximum (hence an absolute ma? maximum) since H”(r) < O on (0,1). When r : k——l—1 H()— x/E\/__ f1 11 [6—1—1 ka+1+1¢k+lzdk+11h1ustheasseltionistrueforn:k:+1 when it is true for n : k. By induction. it is true for all positive integers n. Note: In general. a maximally tall stack of n hemispherical bubbles consists of bubbles with — g 2
radi11.1/:l—l.1/n {WM—Ml.
’I’L TL Tl n ...
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 Spring '10
 Ban
 Calculus

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