anonymous
  • anonymous
please help me out with this one- a right circular cone has a surface area of 718 square inches. what dimensions (radius and height) will result in a maximum volume?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
I began the problem, and now it after 30 mins of work it is a nightmare.
Mertsj
  • Mertsj
What class is this for--calculus?
anonymous
  • anonymous
yup

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anonymous
  • anonymous
so far, i've taken the SA eqn and tried to optimize it with substitution, but then after finding a derivative it just is bad.
Mertsj
  • Mertsj
If you could find satellite of myininaya they could help you
amistre64
  • amistre64
this one, jwg lol
Mertsj
  • Mertsj
yep
amistre64
  • amistre64
is there a formula for the surface area of a rt circular cone?
Mertsj
  • Mertsj
SA=pir^2+pirl
amistre64
  • amistre64
and perferably one for volume as well to relate the 2?
amistre64
  • amistre64
1/3 something is all i can remember
amistre64
  • amistre64
1/3 volume of a cylinder perhaps?
Mertsj
  • Mertsj
V=-1/3pir^2h
Mertsj
  • Mertsj
Exactly
amistre64
  • amistre64
ok; looks like we are given an value for SA so that we can find an r perhaps?
Mertsj
  • Mertsj
The problem is the slant height adds a variable.
amistre64
  • amistre64
i take it thats the "l" part in the SA
Mertsj
  • Mertsj
yes
amistre64
  • amistre64
|dw:1327287244737:dw|
Mertsj
  • Mertsj
precisely
amistre64
  • amistre64
l = sqrt(h^2 + r^2) then 718 = pi r^2 + pi r(sqrt(h^2+r^2)) 718/pir = r + (sqrt(h^2+r^2)) (718/pir) - r = sqrt(h^2+r^2) [(718/pir) - r]^2 = h^2+r^2 [(718/pir) - r]^2 - r^2 = h^2 i wonder of thats going according to plan :)
amistre64
  • amistre64
at any rate we can define h in terms of r; aint pretty but doable
Mertsj
  • Mertsj
That should be progress. Wonder where the asker is.
amistre64
  • amistre64
fetal position, sobbing maybe lol
amistre64
  • amistre64
V = 1/3 pi r^2 h \[V=\frac{pi\ r^2\ \sqrt{(\frac{718}{pi\ r}-r)^2-r^2}}{3}\] looks about right?
Mertsj
  • Mertsj
yes
Mertsj
  • Mertsj
Now I suppose we have to differentiate that and set to 0?
amistre64
  • amistre64
the 1/3 pi is a constant that can be put to the side and we are left with deriving:\[V=r^2\ ((\frac{718}{pi\ r}-r)^2-r^2)^{1/2}\]
amistre64
  • amistre64
yep, find the zeros and the undefineds to test
amistre64
  • amistre64
\[V'=r'^2\ ((\frac{718}{pi\ r}-r)^2-r^2)^{1/2}+r^2\ ((\frac{718}{pi\ r}-r)^2-r^2)'^{1/2}\] \[V'=2r\ ((\frac{718}{pi\ r}-r)^2-r^2)^{1/2}+\frac{1}{2}r^2\ ((\frac{718}{pi\ r}-r)^2-r^2)^{-1/2}*((\frac{718}{pi\ r}-r)^2-r^2)'\] \[V'=2r\ ((\frac{718}{pi\ r}-r)^2-r^2)^{1/2}+\frac{1}{2}r^2\ ((\frac{718}{pi\ r}-r)^2-r^2)^{-1/2}*(\frac{718}{pi\ r}-r)'^2-r'^2\] \[V'=2r\ ((\frac{718}{pi\ r}-r)^2-r^2)^{1/2}+\frac{1}{2}r^2\ ((\frac{718}{pi\ r}-r)^2-r^2)^{-1/2}*2(\frac{718}{pi\ r}-r)(\frac{718}{pi\ r}-r)'-2r\] yep, its a doozie; the wolf might be quicker
Mertsj
  • Mertsj
That's what I did.
amistre64
  • amistre64
..... slacker!! lol
Mertsj
  • Mertsj
proudly so
Mertsj
  • Mertsj
r=7.56
amistre64
  • amistre64
yeah, i was gonna say around 8
amistre64
  • amistre64
plug that value in to get the V max
Mertsj
  • Mertsj
I just ignored the miserable denominator and set the numerator to 0.
Mertsj
  • Mertsj
Must first find the height
amistre64
  • amistre64
[(718/pir) - r]^2 - r^2 = h^2 plug in yer r
Mertsj
  • Mertsj
yep
amistre64
  • amistre64
i dunno if solving in terms of h would have been easier
Mertsj
  • Mertsj
tooooooooooooooooooooooooooooo late
amistre64
  • amistre64
lol, since h was buried i assume not
Mertsj
  • Mertsj
h= 21.37
amistre64
  • amistre64
that should do it then
Mertsj
  • Mertsj
Except for checking to make sure those values result in the correct surface area.
amistre64
  • amistre64
it has to, we did everything correctly
Mertsj
  • Mertsj
Of course...no mistakes...ever!!
amistre64
  • amistre64
they wanted max volume
amistre64
  • amistre64
the wolf helped so we can always pass the blame ;)
Mertsj
  • Mertsj
working on that.
Mertsj
  • Mertsj
1279.02= max volume
Mertsj
  • Mertsj
Good to have a scapegoat.
Mertsj
  • Mertsj
Hey!! It's right. SA = 717.9758
Mertsj
  • Mertsj
Thank you very much.

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