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Expert: Alon Mandes - 7/23/2009


Question
A spherical ball of wax melts away such that its volume decreases at a rate
proportional to its surface area. If r(t) is its radius, show dR/dt equals some
constant (negative). If, at time t=t2 the wax is down to half its original volume,
at what time does it vanish all together? (Your expression will involve the
constant t2)

Answer
V(t)=(4/3)πr³(t) & S(t)=4πr²(t). Where S is the surface area.
We know that dV/dt=-αS(t) . That means :
4πr²(t)(dr/dt)=-α4πr²(t). Therefore, dr/dt=-α .
Let's write an expression for r(t) as function of time :
r(t)=∫r'(t)dt=ro-αt. Where ro is the initial radius.
At time t2 : V(t2)=½Vo -> r(t2)=½ro : ro-αt2=½ro -> ro=2αt2 .
The ball vanish when r(t)=0 , let's denote this time as tf :
r(tf)=0 -> ro-αtf=0 -> tf=ro/α=2t2 .
This result is highly expected since the radius is changing linearly in time !  ("there is no ODE here!")

Alon.

Calculus

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

Experience

1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

Education/Credentials
M.A in Mathematics & Bs.c in Electronics.

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