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In order to minimize neutron leakage from a reactor, the ratio of the surface area to the volume must be as small as possible. Assume that a sphere of radius a and a cube both have the same volume. Find the surface - to - volume ratio for (a) the sphere and (b) the cube. (c) Which of these reactor shapes would have the minimum leakage?

a) $4.83\mathrm{V}^{-1/3}$

b) $6\mathrm{V}^{-1/3}$

c) The sphere would have minimum leakage

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Cornell University

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University of Washington

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for discussion. We are given debt for the rector to minimize new 20 kitsch. We want to be mined the surface area to volume ratio which is so face area folio. I want you to be s small s possible, right? So now we're considering two different ships over here. First is this fear And now we're assuming that both off them have t seem followem. Now we want to find what the surface to volume ratio and compare between them We would have to take the ratio and in terms off the out for you. Right, So starting for Utd Sphere, the surface area Facey really escape on this for pie are square Defoe William is for hi are cute in order to convert to convert the surface area to volume ratio to only depend on volume we will have to make the surface area depend only on for you. So from people you here we can find it R equals two You fee over for pie to the pope wanted We substitute that and you get no Sophie. See where to be for by street view for pie to oppose. It hurts all right. And so surveys Europa volume given s for pay times. True for for pie to pull off to test times V off. Negative. Wanted. Next we look for you. Okay, Now we're looking at a cube that say yes. He length off l so surface area would be given s l square times six. It seems there are six sites off the cute for the volume. Just simply el que. And from here, we can rearrange to find L. A. To be close to veto power. Substituting this into surface area. You get a sec close to six times veto oppo off. That's no que get e ratio. They renew and it is six times V to the power off. Negative one. Now we can compare right? The constant that is behind our veto power wanted, since both of them have this Ah, variable veto power wanted. So what we have to do is just compare this constant value. Your we simply have to put it into our calculator. So 40 please feel you should gets value, which you can put into the Graito. I'll just give me a moment. This is four point It's tree V to the post. Negative one. And so by comparing this constant over here, we see that this fear is actually lower. Smaller than six times feet, about one right. And so the surface area to volume ratio is smaller. 40. This compared to a T.

National University of Singapore