Why not?
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!)
$\endgroup$ 11 Answer
$\begingroup$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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