A student was studying for his examinations and his lights went off.it was around 1:00am.he lighted two uniform candles of equal lenght but one thicker than the other.the thick candle is supposed to last 6 hours and thin one illuminates for 4 hours.when he finally went to sleep,the thich candle was twice as long as the thin one.for how long did the student study in candle light?




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Let us assume that a candle burns equal amounts of mass in equal durations of time and gives uniform amount of light throughout.  This assumption can be made as the wick size determines the mass burnt.  Let us assume that the densities "d" of both candles is same.  Given heights (lengths) "h" are same.

      mass of candle 1 = π R₁² h d,            R is the radius of the cylindrical candle
          mass of candle 2 = π R₂² h d,

Hence, the total time duration for the burning of candles are proportional to mass

                 R₁² : R₂² = 6 : 4         --- equation 1
         After some time t, the masses are :

remaining mass of candle1 =  π R₁² h₁ d 
remaining mass of candle 2  = π R₂² h₂ d  ,        2 h₂ = h₁

Amount of wax burnt is same in both cases.

          π R₁² h d - 2 π R₁² h₂ d = π R₂² d h - π R²₂ h₂ d
          R₁² (h - 2 h₂)  = R₂² (h - h₂) ,         now using equation 1, 
            6 (h - 2 h₂) =  h  - h₂

           5 h = 11 h₂

Hence, candle 2,  became 5/11 th of its length/height.  The length consumed is 6/11 of h.

        Hence the time during which the student studied under the candle light is
                    6/11 * 4 hours    = 24/11 hours = 2.189 hours 

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