The sum of the surfaces of a sphere of radius x and a cube of side y is always a constant k . if V is the sum of there volumes ,show that V is least when y=2x
2006-08-31 02:33:10 · 12 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
After a lot of arithmetic mistakes I got it. And it looks like
I'm the first judging from all the nonsense before me.
The sum of the areas of sphere and cube being a constant is expressed as
1) 4pix^2+6y^2=k, the sum of the volumes is
2) V=(4/3)pix^3 +y^3
In 1) solve for y and it's y=((4pix^2-k)/(-6))^(1/2) and
substitute this for y in 2) getting
3) V=(4/3)(pix^3)+
((4pix^2-k)/(-6))^(3/2)
V is minimum when dV/dx = 0 and taking the derivative of
3) w/respect to x we have
dV/dx=4pix^2+
(3/2)((4pix^2-k)/(-6))^(1/2)
(4pix/(-3)).
Sorry I have to split the lines like this.
But they get chopped off by the yahoo editor.
Now set dV/dx to 0, bring 4pix^2 to the left side of the equation and square both sides. That gets rid of that
pesky factor to the (1/2) power. Now we have
16pi^2x^4=(9/4)
{(4pix^2-k)/(-6)}
(16pi^2x^2)/9 which reduces to
5) x^2+pix^2/6=k/24 (That's where I screwed up a lot). Now solve for k.
6) k=24x^2+4pix^2 and set this equal to k in equation 1)
This gives us 24x^2+4pix^2=4pix^2+6y^2 and simplifying, it's 4x^2=y^2 and finally
2x=y . Ta-Dahhh!
2006-09-01 02:56:57 · answer #1 · answered by albert 5 · 0⤊ 0⤋
well I didn't quitte get u but I tryed to do something with the only information u gave us,so let's name the surface of the sphere with As and the one of the cube with Ac,so I think that some of the information u gave us are:
As+Ac=k
V=Vs+Vc
y=2x and then x=y/2
so u want us to find out if V is least when y=2x?
all I know is that:
As=4piR*=4pi x*(*means 2)
R*=x (your radius x)
Ac=6y*
Ac=6(2x)*=24x* which is also=24(y/2)*=6y*
but I don't know why we need the surfaces when we gotta demonstrate that V is at least when y=2x?
Vs=4pi x* (now * is 3)
Vc=y*=(2x)*
now these are just formulas but I think the demonstration should be:
V=Vs+Vc=(4pi x*)/3+8x* and V must be at least when y=2x??????
now will u excuse me a gotta get some medicines for the next 24 hours 'cause I have a headake!!! jocking!
man I was brave,I tryed hard,though I didn't quitte got what u wanted to find out,sorry for this but I hope u'll finally get your answer somehow!goodluck!plsplspls give me the 10 for the effort for my headake,he he!
2006-08-31 07:03:00 · answer #2 · answered by denise p 4 · 0⤊ 0⤋
trick question.
the radius of the sphere has to be half of the length of the sid for the sphere to fit in the cube and give both as the smallest volume
2006-08-31 02:40:50 · answer #3 · answered by squashantha 2 · 0⤊ 0⤋
Get the volumes in terms of x and y and then differentiate V interms of x. You will get the result.
And yes I am brave, really brave.
2006-08-31 02:45:04 · answer #4 · answered by A 4 · 0⤊ 0⤋
how is bravery related to volume of sphere and area of surface. do u always go to gym or is ur house next to gym.
2006-08-31 16:23:15 · answer #5 · answered by dilipagr_2000 2 · 0⤊ 0⤋
I may be brave,but l am not necessarily smart.
2006-08-31 02:40:49 · answer #6 · answered by Anonymous · 0⤊ 1⤋
Note very. Do your homework on your own dude !!
2006-08-31 02:40:15 · answer #7 · answered by sabeesh 2 · 0⤊ 1⤋
Not nearly that brave (or bright)
2006-08-31 02:38:47 · answer #8 · answered by corpuscollossus 3 · 0⤊ 1⤋
very good way of doin one's homework.
2006-08-31 02:57:46 · answer #9 · answered by Anonymous · 0⤊ 0⤋
Are you asking someone to do your homework?
2006-08-31 02:40:22 · answer #10 · answered by Anonymous · 0⤊ 1⤋