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I have an ellipse with area $\pi ab$ $a = 6$, $b = 4$ these are the axis lengths. I am suppose to compute the volume of a cone of height 12.

I tried many solutions but none of them worked and I don't know why. I would type them up but I doubt much could be learned. Basically I have been trying to use the fact that $\pi ab$ is the area that I can find $a$ or $b$ and then that ratio is my radius at any point. This doesn't really work though and I don't know why.

How do I do this?

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    $\begingroup$ Have you looked at the section Volume in Wikipedia article Cone? $\endgroup$ – Zev Chonoles Jun 27 '13 at 17:03
  • $\begingroup$ No but I think I will be able to in four years of intensive math study. $\endgroup$ – user84004 Jun 27 '13 at 19:45
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    $\begingroup$ What solutions have you tried? Where did you get stuck? $\endgroup$ – Code-Guru Jun 27 '13 at 20:42
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Assume the cone is vertical, its axis is the $z$-axis and its base is an ellipse on the $xy$-plane, centered at $(x,y,z)=(0,0,0)$, with semi-major axis $a$ and semi-minor axis $b$. The equation of the base is thus

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\qquad z=0.$$

Let $h$ be the height of the cone. The cone cross section at height $z$ is an ellipse with semi-major axis $$x_1=a\left( 1-\frac{z}{h}\right) $$ and semi-minor axis $$x_2=b\left( 1-\frac{z}{h}\right) $$ by similarity of (right) triangles, as shown in the following sketch:

enter image description here

The equation of the boundary of this cross section is

$$\frac{x^2}{x_1^2}+\frac{y^2}{x_2^2}=1,\qquad z=z.$$

Basically I have been trying to use the fact that $\pi ab$ is the area

The area $A(z)$ of this cross section is thus $$ \begin{equation*} A(z)=\pi x_1 x_2=\pi ab\left( 1-\frac{z}{h}\right) ^{2}. \end{equation*} $$

The volume is the integral of the area $A(z)$ from $z=0$ to $z=h$

$$ \begin{eqnarray*} V &=&\int_{0}^{h}A(z)\, dz=\int_{0}^{h}\pi ab\left( 1-\frac{z}{h}\right) ^{2}dz \\ &=&\pi ab\int_{0}^{h}\left( 1-2\frac{z}{h}+\frac{z^{2}}{h^{2}}\right) dz \\ &=&\pi ab\left( h-h+\frac{h}{3}\right) =\frac{h}{3}\pi ab, \end{eqnarray*} $$ i.e. $$V=\frac{1}{3}A_{\text{base}}\times \text{height},$$

as expected. For $a=6,b=4,h=12$, we have: $$ \begin{equation*} V=\frac{h}{3}\pi ab=96\pi. \end{equation*} $$

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  • $\begingroup$ I don't follow your area formula. The cross section is an ellipse is it not? $\endgroup$ – user84004 Jun 27 '13 at 19:41
  • $\begingroup$ @user84004 Yes, it is an ellipse but with semi-major axis $x_1$ and semi-minor axis $x_2$. For $z=0$ the cross section is just the base, an ellipse with semi-major axis $a$ and semi-minor axis $b$. For $z$ variable $x_1$ and $x_2$ are functions of $z$ as written above: $$x_1=a\left( 1-\frac{z}{h}\right) $$ and $$x_2=b\left( 1-\frac{z}{h}\right) .$$ $\endgroup$ – Américo Tavares Jun 27 '13 at 19:46
  • $\begingroup$ I need to back up, I don't understand how you are getting a and b. What is 1 - z/h? $\endgroup$ – user84004 Jun 27 '13 at 19:48
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    $\begingroup$ I see that now, this seems like an immensely complicated problem. I will try and do it from the start with your graph. $\endgroup$ – user84004 Jun 27 '13 at 20:39
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    $\begingroup$ Thank you that makes sense. $\endgroup$ – user84004 Jun 27 '13 at 23:01
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Hint: Assume the cone is tip up. By similarity, the area of horizontal cross-section at distance $w$ down from the apex of the cone is $$\left(\frac{w}{12} \right)^2 (\pi ab).$$ Integrate, $w=0$ to $12$.

Better, imagine that the cone is tip down, with tip at the origin. Then the area of cross-section a distance $z$ up from the $x$=$y$ plane is $(z/12)^2(\pi ab)$. For the volume, integrate from $z=0$ to $z=12$.

If you prefer (I don't) you can have the cone in conventional position, and measure distance up from the $x$-$y$ plane. Then the area of cross-section at distance $z$ above the $x$-$y$ plane is $\left( \frac{12-z}{12}\right)^2 (\pi ab)$.

Added: For the integration, we want $$\int_0^{12} (\pi ab)\frac{w^2}{12^2}\,dw.$$ Calculation gives $(\pi ab)\frac{(12)^3}{3\cdot 12^2}$.

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  • $\begingroup$ This doesn't seem to fit with what I know and with what Zeev linked: the volume is one third of the base's area times the height, and in this case it is $$\frac{12\pi ab}3=4\pi ab\;...$$ $\endgroup$ – DonAntonio Jun 27 '13 at 17:14
  • $\begingroup$ I don't understand where w/12 comes from and why it is squared. $\endgroup$ – user84004 Jun 27 '13 at 17:16
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    $\begingroup$ @user84004: By similarity, the cross-section $w$ down from the tip is similar to the base of the cone, but shrunk by a factor $w/12$. So its area is shrunk by a factor $(w/12)^2$. $\endgroup$ – André Nicolas Jun 27 '13 at 17:20
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    $\begingroup$ When you scale the linear dimensions of an object by the scaling factor $r$, then the area is scaled by the factor $r^2$. For example, halfway along, the area of cross-section of the ellipse is $(1/2)^2$ times the area of the big ellipse at the base. $\endgroup$ – André Nicolas Jun 27 '13 at 17:44
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    $\begingroup$ An ellipse has two radii, but let's use the word loosely. The scaling factor is the ratio of the radius of the ellipse of cross-section and the radius of the ellipse which which is the base. So as I mentioned earlier, if we take the cross-section halfway, the ellipse of cross-section has radius half the base ellipse, so the scaling factor is $1/2$, or maybe more clearly $6/12$. There is now a nice answer with picture that explains everything in detail. $\endgroup$ – André Nicolas Jun 27 '13 at 19:43

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