show that the height of the cylinder of maximum volume that can be inscribed within a cone of height $h$ is...
$begingroup$
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
104k1468120
asked Mar 19 '17 at 12:40
Dhruv RaghunathDhruv Raghunath
156412
$endgroup$
add a comment |
$begingroup$
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
104k1468120
asked Mar 19 '17 at 12:40
Dhruv RaghunathDhruv Raghunath
156412
$endgroup$
$begingroup$
See this for finding the maximum volume of a cylinder in a cone of height $h$.
$endgroup$
– heather
Mar 19 '17 at 13:03
add a comment |
$begingroup$
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
104k1468120
asked Mar 19 '17 at 12:40
Dhruv RaghunathDhruv Raghunath
156412
$endgroup$
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
104k1468120
asked Mar 19 '17 at 12:40
Dhruv RaghunathDhruv Raghunath
156412
edited Nov 27 '17 at 21:02
Siong Thye Goh
104k1468120
asked Mar 19 '17 at 12:40
Dhruv RaghunathDhruv Raghunath
156412
edited Nov 27 '17 at 21:02
Siong Thye Goh
104k1468120
edited Nov 27 '17 at 21:02
Siong Thye Goh
104k1468120
edited Nov 27 '17 at 21:02
Siong Thye Goh
104k1468120
104k1468120
asked Mar 19 '17 at 12:40
Dhruv RaghunathDhruv Raghunath
156412
asked Mar 19 '17 at 12:40
Dhruv RaghunathDhruv Raghunath
156412
asked Mar 19 '17 at 12:40
Dhruv RaghunathDhruv Raghunath
156412
156412
$begingroup$
See this for finding the maximum volume of a cylinder in a cone of height $h$.
$endgroup$
– heather
Mar 19 '17 at 13:03
add a comment |
$begingroup$
See this for finding the maximum volume of a cylinder in a cone of height $h$.
$endgroup$
– heather
Mar 19 '17 at 13:03
$begingroup$
See this for finding the maximum volume of a cylinder in a cone of height $h$.
$endgroup$
– heather
Mar 19 '17 at 13:03
$begingroup$
See this for finding the maximum volume of a cylinder in a cone of height $h$.
$endgroup$
– heather
Mar 19 '17 at 13:03
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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 NovatiEmilio Novati
52.2k43574
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
oldest
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oldest
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active
oldest
votes
$begingroup$
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 NovatiEmilio Novati
52.2k43574
$endgroup$
add a comment |
$begingroup$
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 NovatiEmilio Novati
52.2k43574
$endgroup$
add a comment |
$begingroup$
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 NovatiEmilio Novati
52.2k43574
$endgroup$
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 $.
Maximize this function.
answered Mar 19 '17 at 13:46
Emilio NovatiEmilio Novati
52.2k43574
answered Mar 19 '17 at 13:46
Emilio NovatiEmilio Novati
52.2k43574
answered Mar 19 '17 at 13:46
Emilio NovatiEmilio Novati
52.2k43574
answered Mar 19 '17 at 13:46
Emilio NovatiEmilio Novati
52.2k43574
52.2k43574
add a comment |
add a comment |
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$begingroup$
See this for finding the maximum volume of a cylinder in a cone of height $h$.
$endgroup$
– heather
Mar 19 '17 at 13:03