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Why not?

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I'm trying to design a calculus I problem. I want them to eventually prove that the volume of a cone (oblique or even with irregular base) is $\frac{\pi r^2 h}{3}$, I also want them to know what a cone is (and what it isn't) -- so I thought this might be a good starting question. But, I don't know if it goes where I want it to.

Update: Making this more clear... I should have said more!

Imagine each "nose cone" has its vertex at the origin and the perpendicular (from the vertex to the base) is the x-axis. If you have a formula A(x) for the cross-sectional area of the "nose cone" at x, perpendicular to the x-axis then the volume is $\int_0^h A(x)dx$.

So, here we have formulas for the volumes of a few solids from NASA. (Only one is a cone.) Can we recover $A(x)$ from these? No, we can't! Not without assuming that the cones are not oblique or otherwise irregular.

(I'll take this as a sign that this would confuse my students!)

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@a little don: I'm sorry, I don't quite understand the question! –  Juan S May 3 '11 at 6:54

1 Answer 1

The cross-sectional area is simply the derivative of the volume with respect to h, by the Fundamental Theorem of Calculus. For a problem, how about a cone whose base is a cardioid?

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You mean with respect to x, right? h is a constant. –  futurebird May 3 '11 at 7:14
@a little don: what is $x$? –  Fabian May 3 '11 at 7:15
The active variable in the integral used to find the volume. It doesn't appear in the final equation... that's the whole point. –  futurebird May 3 '11 at 7:19
@a little don: $V'=A$, no matter if you write it as $V'(x)=A(x)$ or $V'(h)=A(h)$... –  Hans Lundmark May 3 '11 at 7:19
a little don, it is not possible to determine the shape of the base, only its area. –  Dan Brumleve May 3 '11 at 7:33

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