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(B) Let the length of the candle be $L$ and its density be $\rho_c$. Let the density of the liquid be $\rho_l$. For the candle to float,the weight of

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fall at the rate of $1 \, cm/hour$

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(B) Let the length of the candle be $L$ and its density be $\rho_c$. Let the density of the liquid be $\rho_l$. For the candle to float,the weight of the candle must be equal to the buoyant force: $A \cdot L \cdot \rho_c \cdot g = A \cdot h \cdot \rho_l \cdot g$,where $h$ is the submerged length and $A$ is the cross-sectional area of the candle. This gives $h = L \cdot (\rho_c / \rho_l)$. Since the candle is floating,the ratio $\rho_c / \rho_l$ is constant. Let $k = \rho_c / \rho_l$. Then $h = k \cdot L$. The height of the top of the candle above the liquid surface is $H = L - h = L - k \cdot L = L(1 - k)$. When the candle burns,its length $L$ decreases at a rate $v = dL/dt = 2 \, cm/hour$. The rate of change of the height of the top of the candle is $dH/dt = (dL/dt) \cdot (1 - k)$. Assuming the candle is made of wax with density approximately half that of water (or generally,for a candle to float,$k Thus,the top of the candle falls at the rate of $1 \, cm/hour$.

$A$ candle of diameter $d$ is floating on a liquid in a cylindrical container of diameter $D$ $(D >> d)$ as shown in the figure. If it is burning at the rate of $2 \, cm/hour$,then the top of the candle will

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