show that the height of the cylinder of maximum volume that can be inscribed within a cone of height $h$ is...
show that the height of the cylinder of maximum volume that can be inscribed within a cone of height $h$ is $frac{h}3$.
I have tried solving this sum but am unable to substitute the radius of the cylinder in terms of $h$.
derivatives maxima-minima
edited Nov 27 '17 at 21:02
Siong Thye Goh
98.9k1464116
asked Mar 19 '17 at 12:40
Dhruv Raghunath
156212
bumped to the homepage by Community♦ yesterday
This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.
add a comment |
show that the height of the cylinder of maximum volume that can be inscribed within a cone of height $h$ is $frac{h}3$.
I have tried solving this sum but am unable to substitute the radius of the cylinder in terms of $h$.
derivatives maxima-minima
edited Nov 27 '17 at 21:02
Siong Thye Goh
98.9k1464116
asked Mar 19 '17 at 12:40
Dhruv Raghunath
156212
bumped to the homepage by Community♦ yesterday
This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.
See this for finding the maximum volume of a cylinder in a cone of height $h$.
– heather
Mar 19 '17 at 13:03
add a comment |
show that the height of the cylinder of maximum volume that can be inscribed within a cone of height $h$ is $frac{h}3$.
I have tried solving this sum but am unable to substitute the radius of the cylinder in terms of $h$.
derivatives maxima-minima
edited Nov 27 '17 at 21:02
Siong Thye Goh
98.9k1464116
asked Mar 19 '17 at 12:40
Dhruv Raghunath
156212
show that the height of the cylinder of maximum volume that can be inscribed within a cone of height $h$ is $frac{h}3$.
I have tried solving this sum but am unable to substitute the radius of the cylinder in terms of $h$.
derivatives maxima-minima
derivatives maxima-minima
edited Nov 27 '17 at 21:02
Siong Thye Goh
98.9k1464116
asked Mar 19 '17 at 12:40
Dhruv Raghunath
156212
edited Nov 27 '17 at 21:02
Siong Thye Goh
98.9k1464116
asked Mar 19 '17 at 12:40
Dhruv Raghunath
156212
edited Nov 27 '17 at 21:02
Siong Thye Goh
98.9k1464116
edited Nov 27 '17 at 21:02
Siong Thye Goh
98.9k1464116
edited Nov 27 '17 at 21:02
Siong Thye Goh
98.9k1464116
98.9k1464116
asked Mar 19 '17 at 12:40
Dhruv Raghunath
156212
asked Mar 19 '17 at 12:40
Dhruv Raghunath
156212
asked Mar 19 '17 at 12:40
Dhruv Raghunath
156212
156212
bumped to the homepage by Community♦ yesterday
This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.
bumped to the homepage by Community♦ yesterday
This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.
See this for finding the maximum volume of a cylinder in a cone of height $h$.
– heather
Mar 19 '17 at 13:03
add a comment |
See this for finding the maximum volume of a cylinder in a cone of height $h$.
– heather
Mar 19 '17 at 13:03
See this for finding the maximum volume of a cylinder in a cone of height $h$.
– heather
Mar 19 '17 at 13:03
See this for finding the maximum volume of a cylinder in a cone of height $h$.
– heather
Mar 19 '17 at 13:03
add a comment |
1 Answer
1
active
oldest
votes
Hint:
Look at the figure. With:
$$
BH=h quad HC=R quad FG=x quad HG=r
$$
from the similarity of the triangles $BHC$ and $FGC$ we have:
$$
h:x=R:(R-r)
$$
so that $r=frac{R}{h}(h-x)$
Now you can express the volume of the cylinder as $V=pi r^2x=pi frac{R^2}{h^2}(h-x)^2x $.
Maximize this function.
answered Mar 19 '17 at 13:46
Emilio Novati
51.5k43472
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2193469%2fshow-that-the-height-of-the-cylinder-of-maximum-volume-that-can-be-inscribed-wit%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hint:
Look at the figure. With:
$$
BH=h quad HC=R quad FG=x quad HG=r
$$
from the similarity of the triangles $BHC$ and $FGC$ we have:
$$
h:x=R:(R-r)
$$
so that $r=frac{R}{h}(h-x)$
Now you can express the volume of the cylinder as $V=pi r^2x=pi frac{R^2}{h^2}(h-x)^2x $.
Maximize this function.
answered Mar 19 '17 at 13:46
Emilio Novati
51.5k43472
add a comment |
Hint:
Look at the figure. With:
$$
BH=h quad HC=R quad FG=x quad HG=r
$$
from the similarity of the triangles $BHC$ and $FGC$ we have:
$$
h:x=R:(R-r)
$$
so that $r=frac{R}{h}(h-x)$
Now you can express the volume of the cylinder as $V=pi r^2x=pi frac{R^2}{h^2}(h-x)^2x $.
Maximize this function.
answered Mar 19 '17 at 13:46
Emilio Novati
51.5k43472
add a comment |
Hint:
Look at the figure. With:
$$
BH=h quad HC=R quad FG=x quad HG=r
$$
from the similarity of the triangles $BHC$ and $FGC$ we have:
$$
h:x=R:(R-r)
$$
so that $r=frac{R}{h}(h-x)$
Now you can express the volume of the cylinder as $V=pi r^2x=pi frac{R^2}{h^2}(h-x)^2x $.
Maximize this function.
answered Mar 19 '17 at 13:46
Emilio Novati
51.5k43472
Hint:
Look at the figure. With:
$$
BH=h quad HC=R quad FG=x quad HG=r
$$
from the similarity of the triangles $BHC$ and $FGC$ we have:
$$
h:x=R:(R-r)
$$
so that $r=frac{R}{h}(h-x)$
Now you can express the volume of the cylinder as $V=pi r^2x=pi frac{R^2}{h^2}(h-x)^2x $.
Maximize this function.
answered Mar 19 '17 at 13:46
Emilio Novati
51.5k43472
answered Mar 19 '17 at 13:46
Emilio Novati
51.5k43472
answered Mar 19 '17 at 13:46
Emilio Novati
51.5k43472
answered Mar 19 '17 at 13:46
Emilio Novati
51.5k43472
51.5k43472
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2193469%2fshow-that-the-height-of-the-cylinder-of-maximum-volume-that-can-be-inscribed-wit%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
See this for finding the maximum volume of a cylinder in a cone of height $h$.
– heather
Mar 19 '17 at 13:03