I understand that I should take the an initial derivative and set it same to zero, i m sorry will find the maximum.

My difficulty is just how to deal with the change $h$, the height?

How deserve to I rewrite that in terms of $x$ or $r$.

I have tried come implicitly identify this equation, but that wasn"t valuable for me.


*

*

The elevation $h$ is the distance from the apex come the circular base. That is in between $0$ and $2r$ (with $r$ the radius the the sphere), and the radius of the base grows together $h$ rises from $0$ to $r$ and then shrinks again as $h$ boosts to $2r$. A diagram might convince you that by Pythagoras $(h-r)^2+x^2=r^2$, where $x$ is the radius that the base. That allows you to eliminate $x^2$ in favour of $2rh-h^2$, therefore you have $V(h)=\frac13(2rh-h^2)h$ and thus $V"(h)=\frac13(4rh-3h^2)$, i beg your pardon vanishes in ~ $h=\frac43r$ and also thus $x=\frac\sqrt83r$, with maximal volume $V=\frac13\pi\frac89r^2\frac43r=\frac3281\pi r^3=\frac827V_\text s$, wherein $V_\text s$ is the volume that the sphere.

You are watching: Find the volume of the largest right circular cone that can be inscribed in a sphere of radius r


re-superstructure
point out
follow
edited Oct 31 "12 at 1:30
reply Oct 31 "12 at 1:24
*

jorikijoriki
210k1414 yellow badges247247 silver- badges452452 bronze badges
$\endgroup$
include a comment |
12
$\begingroup$
*

Ok. In the picture above, there are two sketches. The one on the left is in 3d, however is kinda difficult to to express to. The one on the right is in 2d and easier to watch what"s walking on, so I"m going to use that (just imagine it"s a cross-section of the 3d pic).

The an enig of the whole difficulty is come relate $h$ and $b$. This is done using the equation because that the right half of the circle: $$x = \sqrtr^2-y^2$$

$b$ is the $x$ value once $y$ is counter from the top of the round by $h$. Thus, the equation in regards to $b$, $h$, and $r$ is: $$b = \sqrtr^2 - (r-h)^2$$

Now to relate to the volume that a cone:$$V_cone = \frac\pi 3 \cdot b^2 h$$$$V_cone = \frac\pi 3 \cdot (r^2 - (r - h)^2) h$$Simplifying:$$V_cone = \frac \pi 3 \cdot (2h^2 r-h^3)$$Differentiate:$$\fracdV_conedh = \frac \pi 3 \cdot (4r - 3h)h$$Solve:$$h = \left\0, \frac4r3\right\$$We desire the maximum, which obviously will occur at the 2nd value of $h$. Finding the maximal volume is a basic algebra difficulty now.


share
cite
follow
edited Jul 13 "15 in ~ 10:38
*

Joonas Ilmavirta
24.5k1010 gold badges5050 silver- badges9292 bronze badges
answer Oct 31 "12 at 1:32
*

apnortonapnorton
16.9k44 gold badges4444 silver badges105105 bronze title
$\endgroup$
include a comment |
3
$\begingroup$
With offered radius r of a round let the inscribed cone have height h climate remaining length without radius is (h-r) permit R it is in radius that cone climate there we obtain a best angle triangle v r as hypotanious R as adjecent and (h-r) together opposite side. Now by pythagorus organize ((h-r)^2)+(R^2)=(r^2). Currently express R in terms of h and also r. We acquire (R^2)=(2hr-hh). Substitute in the volume of cone equation V=(pi/3)(R^2)h. Now V=(pi/3)(2hr-hh)h. Identify wrt h v r as consistent because radius can not be variable and also substitute come zero to gain maxima. We acquire h=(4r/3).


re-publishing
point out
follow
answered Aug 13 "15 in ~ 10:36
SteveSteve
3111 bronze badge
$\endgroup$
1
include a comment |

your Answer


Thanks because that contributing an answer to rwandachamber.orgematics ridge Exchange!

Please be certain to answer the question. Administer details and share her research!

But avoid

Asking because that help, clarification, or responding to various other answers.Making statements based upon opinion; back them increase with recommendations or an individual experience.

Use rwandachamber.orgJax to style equations. Rwandachamber.orgJax reference.

To discover more, view our advice on writing an excellent answers.

See more: How To Use Veinminer In Skyfactory 3, Skyfactory 3 Veinminer


Draft saved
Draft discarded

Sign increase or log in in


sign up making use of Google
authorize up using Facebook
authorize up using Email and also Password
send

Post together a guest


name
email Required, but never shown


Post as a guest


name
email

Required, but never shown


write-up Your answer Discard

By clicking “Post her Answer”, friend agree to our terms of service, privacy policy and also cookie plan


Not the prize you're looking for? Browse other questions tagged calculus geometry or ask your own question.


Featured on Meta
Linked
0
Using Lagrange multipliers to discover volume of a cone
2
What is the best volume of a cylinder that deserve to fit in a sphere of a constant radius?
2
A cone enrolled in a sphere
0
Solving: "What is the largest volume that a appropriate circular cone that can be enrolled in a ball of radius r?" there is no calculus?
connected
0
Calculating the volume of an tilt ellipse cone
0
Minimum Volume that a circular, best cone, with a sphere inscribed in it.
0
detect the volume of one cone.
2
recognize the size of the the appropriate circular cylinder of best volume
1
What are the size of a cylinder enrolled in a ideal circular cone of elevation 4 and base radius 6 v maximum volume?
0
Largest feasible right one cone within a ball
1
biggest cone that deserve to be enrolled in a round
1
Calculus Made basic Exercises IX concern 8(a): maximize volume the cylinder inscriptions in a cone
3
walk the radius the the base equal the height in a appropriate circular cone?
hot Network questions an ext hot concerns

concern feed
subscribe to RSS
concern feed To i ordered it to this RSS feed, copy and paste this URL right into your RSS reader.


rwandachamber.orgematics
firm
stack Exchange Network
site design / logo © 2021 ridge Exchange Inc; user contributions license is granted under cc by-sa. Rev2021.9.24.40305


rwandachamber.orgematics stack Exchange works finest with JavaScript permitted
*

your privacy

By click “Accept every cookies”, you agree ridge Exchange can store cookies on your device and disclose information in accordance through our Cookie Policy.