So this question start asking, why doesn't freezing accord to the entire volume off a lake? So in this situation, the air at both the lake is below zero, and we need that out lake to be at zero degrees and has had enough energy to transform into ice. And because hit must be conducted from the water to cool, cool it to zero degrees. And because of face transition, the entire full volume off the water is not that face transition temperature. So on Lee, they they're a top part here, can perform in tow, tow ice. And that's because that's why the whole border for doesn't transform because you can propagate in left and there's you through the ice to make this part off the water to lose enough energy to tow toe past it. Face transition becomes our become ice now, in part to be, they ask you, ah, to derive a relation between the thickness off the eyes and the time it take toe to form this eyes. So let's walk to how we can do that. So the heat that must leave the water in the order to freeze must pass to this to tow this block of ice. So the larger this block is, the harder the hit who have to lower the heat. But they have to travel and worst, this process will become. Let's go see the section off the ice that has area cross sectional area A. So this Here we'll be the cross sectional area A and let at time t This has a height off hate. So eight as a friend Charity in a few days, more interval. So the tea we have, I think there's more sickness that it's the age and the mass off This infant is more block. Well, it will be. The M is equal to rule B V. We assume this density to be constant. So does this row. The cross sectional area is the same. The only thing that gross is this d h here. So this is a relation between the how much mass we have in a I feel it is more block here with the haIf off the block. This is just, uh, Cube. No, the heat must be conducted away from this mess off war to phrase it. So the Q is equal to the end. Well, if and we have This infant is more mass. LF is a lot of heat off the water there it is necessary to to transform Ah ah, War to toe ice. So this gives role. Hey! Oh, uh, the age. So here we have a relation between the heat and the height Now that we know how to ride the change of hate. So h is the change of heat is you could thank you. The t he hears time. This is ego too. The coefficient off the border the off the ice because the heat is propriety in the ice. So Ki I the cross sectional area the difference in temperature 30 age Manus called divided by page. That is the thickness off this block here So we can write this here Don't as the Q is equal to okay, I a the age minus the code times did he divided by age and we can't see that both off these expressions are for Dickie the Cube. And then we can just equal this true toa have a relation beating the H and ditty and integrating this relation. We've got ah, expression that relates the height to the time it takes to generate this this block. So let's do this. We can it quit both sides, so rule a Well, uh, the age is you go to K I A tea hot minus t code Bt over height This thes time we moved to apply in the left side in both sides by age toe pass this guy's toe here and integrate So let's right to they grow from zero to age It's the age physical Tok the age honesty cold rule l f interval from zero to t from the tea and here in the boundary condition at zero height we are at the time That's why both are zero here This is just AIDS square over too. And this is just a And because this hate square virtue will pass this to brother decided take square root So this gives us age is equal to the square old off two. Okay, I the heart dynasty code divided by row l f time screwed off time. So this is their relation that they want to tow Obtain that the the height is proportional to the square Root off fine at part. See, they want to tow to use his equation assuming that the difference in temperature is 10 degrees and from zero to Afghanistan and how much time it takes to form sheets that is 25 centimetres thick, so they want a five time. So to find time, we just, ah, square both sides and passed this so the other side, So equally we can write this equation, Steve, because eight square rule, huh? Divided by Chiu. Okay, I the hut minus c cold and let's play values that they give us so they want a sickness off 0.25. This is the death tee off ice and this is the Latin hit 3 to 4 times. All right, 24 times 10 to the Cube and Jew Times. They hit capacity of the ice and the difference in temperature zero minus manage. Stan is this equal to six points? There's to the fifth seconds. This is 107. The hours now for the last part, they want to know Ah, if the lake is full, different frozen. How much time it pigs, given that the latest foreign mirror sleep. So let's write the equation again. The is eight square root divided by two Hey, I the hot, minus T code. We assume the same difference in temperature and everything else the same. The only difference is this value here for square times 20 times 334 times. Man to the cube, divided by two times 1.6. I'm zero minus. Manage, Stan. And this is absurd number. This is 1.5 times tend to the 10 seconds. This is roughly 500 years, so it will not happen.