Biggest Cuboid in a Sphere
Date: 03/20/2003 at 07:35:21 From: Mitsy Subject: What is the biggest cuboid which fits inside a sphere r=10cm I have a sphere with radius of 10cm. I have to find the biggest cuboid that fits in that sphere. I don't know how I can relate the width, the length and the height of the cuboid. I can find the length and the height, but I don't know how to find the width. For my length and height of the cuboid in the sphere, I drew a cross- sectional diagram of a cuboid in a sphere. It looks like a square in a circle. The diametre of the circle (the sphere) is equivalent to the diagonal of the square (the cuboid) in the diagram, so I could use the Pythagorean theorem to find the length and the height. The sum of the squares of length and height must not go over 400, as the diametre of the sphere is 20, and the square of 20 is 400. But then, there is a problem. Even if I tried different lengths and heights, I would still have no idea of what sort of width the cuboid would have. Is there a formula that could help me to find the width of the cuboid, without knowing the volume, but knowing the length and the height? Is there a formula that relates the length, height and width of a cuboid, without the need to know the volume or the cuboid? Mitsy
Date: 03/21/2003 at 09:38:51 From: Doctor Ian Subject: Re: What is the biggest cuboid which fits inside a sphere r= 10cm Hi Mitsy, This is really just the same problem in three dimensions that you already solved in two dimensions. The largest rectangle that you can fit in a circle is a square. Similarly, the largest cuboid that you can fit in a sphere is a cube. So if you have a cube with side length L, you can use the Pythagorean theorem (or just the formula for the distance between two points, if you've learned that) to show that the distance from the center of the cube to any corner will be ____________________________ D = \| (L/2)^2 + (L/2)^2 + (L/2)^2 And since the cube will touch the sphere at its corners, this is just the radius of the sphere. In this case, you know D, so you need to solve for L. Does that make sense? So, how do you know that there isn't a bigger cuboid, one that isn't a cube, that will fit? This is easier to see in two dimensions, but the idea works in three dimensions, too. Suppose we have a square, with side length L. The area of the square is A = L * L and the distance D from the center to any corner is given by D^2 = (L/2)^2 + (L/2)^2 = 2 * (L/2)^2 But D is the radius of the smallest circle that will fit around the square. Now, suppose we keep the area the same, but make one of the sides longer. The other side has to get shorter, right? If we increase the first side by a factor of k, we have to decrease the other by the same factor. Now we have A = (kL) * (L/k) = L * L * (k/k) So the area is fixed. But what is the distance from the center to any corner? It's now given by D^2 = ((kL)/2)^2 + ((L/k)/2)^2 = k^2(L/2)^2 + (1/k)^2(L/2)^2 = (k^2 + (1/k)^2) * (L/2)^2 How does this compare to the old distance? Since we made one side longer, we know that k > 1. I'll leave it as an exercise for you to show that for k > 1, (k^2 + (1/k)^2) > 2 which means that as we change the square into a rectangle with the same area, the distance from the center to any corner gets larger, so the smallest surrounding circle must get larger. So for a given area, the rectangle with the smallest surrounding circle is a square. You can use the same argument (although it's a little messier) to show that the cuboid with the smallest surrounding sphere is a cube. Which means that the largest cuboid for any given sphere must be a cube. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
Date: 03/22/2003 at 14:54:49 From: Mitsy Subject: Thank you (What is the biggest cuboid which fits inside a sphere r=10cm) Doctor Ian, Thank you SO much for helping me; it was VERY kind of you! I've worked them out, and they all work perfectly. Your explanations were really clear and were very easy to understand. I'd like to thank you again, for bearing with someone like me. It was _really_ kind of you. Mitsy
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