Volume of a SphereDate: 05/19/2002 at 21:58:53 From: Maeve Mungovan Subject: The volume of a sphere I know this topic has already been dealt with previously, and I began to understand it, but you have ommitted some vital information such as: Where does the 4 and 3 come from in (4/3)pi r^3? These issues have not been clearly treated. I have a math exam coming up in a week and it is extremely important that my question is answered. I do not wish to be short, but I am very stressed and your quick response would be much appreciated. Thank you for your time and patience. Maeve Date: 05/19/2002 at 23:18:35 From: Doctor Ian Subject: Re: The volume of a sphere Hi Maeve, I don't wish to be cryptic, but 'why' questions can be tricky. It's not always clear to the person answering the question exactly what counts as an answer to the person asking the question. That's certainly the case here. One way to answer your question is: If you slice a sphere up into very small pieces (using calculus), compute the volumes of the pieces, and add up those volumes, you come out with v = (4/3) pi r^3 But it sounds as though that's not something you would consider a 'proper' answer. So let's try thinking of it another way. Imagine a sphere of radius r. The smallest cube that it would fit around the sphere would have an edge length of 2r, right? So the volume of that cube would be v = (2r)^3 = 8r^3 On the other hand, the largest cube that would fit inside the sphere would have a diagonal of 2r, which means that sqrt(3) L = 2r L = 2r / sqrt(3) where L is the edge length of the cube. So the volume of that cube would be v = (2r/sqrt(3))^3 8r^3 = --------- 3 sqrt(3) 8 sqrt(3) = --------- r^3 9 So if the volume of a sphere is v = something * r^3 it must be the case that 8 sqrt(3) --------- < something < 8 9 1.54 < something < 8 and the actual value, something = (4/3) pi = 4.19 is just about halfway in between. Now, this doesn't say exactly why the coefficient of r^3 should be exactly (4/3)pi, but it does show that it's a plausible value. But you might not accept this as an answer either. If neither of these explanations is what you're looking for, could you please write back and try asking the question in a way that makes it clear what you mean by 'why'? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 05/25/2002 at 14:23:25 From: Maeve Mungovan Subject: The volume of a sphere Actually, your response came just in time, and was greatly appreciated by both me and my younger sister. Thank you, your explanation was clear, precise and simple. Now I will be able to retain the formula for my upcoming examination. Thanks again, Maeve |
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