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(B) Let $m$ be the mass of ice that melts and the mass of water that evaporates per unit time.The heat flow rate required to melt ice at $A$ is $H_1 =

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$9$

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(B) Let $m$ be the mass of ice that melts and the mass of water that evaporates per unit time.The heat flow rate required to melt ice at $A$ is $H_1 = m 	imes L_f = m 	imes 80$.The heat flow rate required to evaporate water at $B$ is $H_2 = m 	imes L_v = m 	imes 540$.Using the formula for heat conduction $H = \frac{KA \Delta T}{L}$,we have:For segment $AP$: $H_1 = \frac{KA(400 - 0)}{\lambda x} = 80m \quad \dots(1)$For segment $PB$: $H_2 = \frac{KA(400 - 100)}{(10 - \lambda)x} = 540m \quad \dots(2)$Dividing equation $(1)$ by equation $(2)$:$\frac{80m}{540m} = \frac{KA(400) / \lambda x}{KA(300) / (10 - \lambda)x}$$\frac{8}{54} = \frac{400}{300} 	imes \frac{10 - \lambda}{\lambda}$$\frac{4}{27} = \frac{4}{3} 	imes \frac{10 - \lambda}{\lambda}$$\frac{1}{9} = \frac{10 - \lambda}{\lambda}$$\lambda = 90 - 9\lambda$$10\lambda = 90$$\lambda = 9$.

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