Find Max Volume of Cylinder Inscribed in Cone

In summary, the formula for finding the maximum volume of a cylinder inscribed in a cone is V = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone. To determine this maximum volume, you need to find the point on the cone's axis where the cylinder's base touches the cone's surface. The volume of the cylinder is directly proportional to the height of the cone, and the maximum volume of a cylinder inscribed in a cone can never be greater than the volume of the cone. Finding this maximum volume is significant in practical applications and helps in understanding the relationship between different geometric shapes and their volumes.
  • #1
Chris L T521
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Thanks again to those who participated in last week's POTW! Here's this week's problem!

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Problem: A right circular cylinder is inscribed in a cone with height $h$ and base radius $r$. Find the largest possible volume of such a cylinder.

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  • #2
This week's question was correctly answered by Ackbach, anemone, MarkFL, and mathworker.

Here's Ackbach's solution (with graphic included by yours truly):

Let the radius of the cylinder be $s$, and its height $g$. The volume of the cylinder is then $V=\pi s^{2}g$. We will put the apex of the cone at the point $(0,h)$, and the center of the base at the origin. Then the center of the cylinder's base will also be at the origin. Consider a section of the cone with the $xy$ plane (see Figure 1). The portion of the section in the first quadrant is a straight line with equation $y=-(h/r)x+h$. Since the cylinder is inscribed, its top must intersect with this section. That is, the point $(s,g)$ must line on this line. Hence, $g=-(h/r)s+h$. Plugging this into the volume formula yields the expression
$$V=\pi s^{2} \left( - \frac{h}{r} s+h\right).$$
Here $V=V(s)$, and $h$ and $r$ are constants. We seek to maximize this function. Therefore, we take its derivative and set it equal to zero to obtain
$$V'(s)=-\frac{3 \pi h s^{2}}{r}+2 \pi h s=0 \implies s=0 \quad \text{or} \quad 2 = \frac{3 s}{r} \implies s= \frac{2r}{3}.$$
Therefore, the maximum volume is
$$V_{ \max}=V(2r/3)=\pi h (2r/3)^{2} \left(1 - \frac{2r/3}{r} \right)=\pi h \cdot \frac{4r^{2}}{27}.$$

Here's anemone's solution:




Let R and H represent the radius and height of the circular cylinder that is inscribed in a cone with height [FONT=MathJax_Math]h[/FONT] and base radius [FONT=MathJax_Math]r[/FONT].

We know that if two triangles are similar, their corresponding side lengths are proportional. We see that the two triangles ACD and BCE are similar, hence,

\(\displaystyle \frac{r}{h}=\frac{R}{h-H}\)

Rearrange the equation to make H the subject (because in this problem we want to find the maximize volume of the cylinder where its formula is \(\displaystyle V_{\text {cylinder}}=\pi R^2h\), it would be easier for us to make the equation of V in terms of only one variable, i.e. to get the formula of H in terms of R.):

\(\displaystyle r(h-H)=rR\)

\(\displaystyle H=\frac{h(r-R)}{r}\)

Hence,

\(\displaystyle V_{\text {cylinder}}=\pi R^2h=\pi R^2\left(\frac{h(r-R)}{r}\right)=\frac{\pi R^2h(r-R)}{r}\)

To find \(\displaystyle V_{\text{max}}\), we find its first derivative w.r.t. R and then use the second derivative test to determine if it's a maximum value:

\(\displaystyle \frac{dV_{\text {cylinder}}}{dR}=\frac{d\left(\frac{\pi R^2h(r-R)}{r}\right)}{dR}=\frac{\pi Rh(2r-3R)}{r}\); \(\displaystyle \frac{d^2V_{\text {cylinder}}}{dR^2}=\frac{2\pi h-6 \pi R}{r}\)

\(\displaystyle \frac{dV_{\text {cylinder}}}{dR}=0\) iff \(\displaystyle 2r-3R=0\;\rightarrow R=\frac{2r}{3}\) \(\displaystyle \rightarrow \frac{d^2V_{\text {cylinder}}}{dR^2}=\frac{2\pi h-6 \pi (\frac{2r}{3})}{r}=-2\pi h (<0)\)

Hence we can conclude now that \(\displaystyle V_{\text{max}}=\pi R^2 h=\pi(\frac{2r}{3})^2(\frac{h(r-R)}{r})=\pi(\frac{4r^2}{9})(\frac{h(r-(\frac{2r}{3}))}{r})=\frac{4 \pi r^2h}{27}\).
 

FAQ: Find Max Volume of Cylinder Inscribed in Cone

What is the formula for finding the maximum volume of a cylinder inscribed in a cone?

The formula for finding the maximum volume of a cylinder inscribed in a cone is V = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone.

How do you determine the maximum volume of a cylinder inscribed in a cone?

To determine the maximum volume of a cylinder inscribed in a cone, you need to find the point on the cone's axis where the cylinder's base touches the cone's surface. This point will be the base of the cylinder and the height of the cylinder will be the same as the height of the cone.

What is the relationship between the volume of the cylinder and the height of the cone?

The volume of the cylinder is directly proportional to the height of the cone. This means that as the height of the cone increases, the volume of the cylinder also increases.

Can the maximum volume of a cylinder inscribed in a cone be greater than the volume of the cone?

No, the maximum volume of a cylinder inscribed in a cone can never be greater than the volume of the cone. This is because the cylinder is inscribed within the cone and cannot have a larger volume than the cone itself.

What is the significance of finding the maximum volume of a cylinder inscribed in a cone?

Finding the maximum volume of a cylinder inscribed in a cone is important in many practical applications, such as designing containers or creating efficient packaging. It also helps in understanding the relationship between different geometric shapes and their volumes.

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