Question

I am willing to give a MEDAL on this one

Answer 1

A. Determine the exact total surface area of a sphere with radius square root of 2 metres

Answer 2

B. an inverted cone with side length 4 metres is placed on top of the sphere so that the centre of its base is 0.5 metres above the centre of the sphere. Find the radius of the cone exactly?

Answer 3

A 11.8432

Answer 4

Most importantly I'm looking for answer B.

Answer 5

In addition A. Is 8Pi metres

Answer 6

surface area of a sphere = 4 pi r^2

Answer 7

Your right on that one

Answer 8

But I'm finding it hard to draw the whole figure

Answer 9

the hint about center of its base allows you to find the vertical height of the cone, so (sqrt 2) - 0.5m

Answer 10

Would that mean that the height of the cone is 0.5m?

Answer 11

How could one draw it?

Answer 12

so you have a side length and you have the vertical length, so use pythagoras theorum to get the radius of the base

Answer 13

will draw now, standby

Answer 14

Thanks a lot jack. Nobody was able to answer this question on this forum. It's been circulating around for two days :)

Answer 15

no, dammit im wrong, sorry, thats not the vertcal height, sorry, is only vertical height to top of sphere, not top of cone, my bad

Answer 16

Is there some other way were we could use trigonometric functions?

Answer 17

What is the chord for?

Answer 18

I've re-read the q, think i have better understanding now
so back to pythag thrm:
r = sqrt 2
h = 0.5 m (above center of sphere)
c = r - h, but pretty much irrelevant for what we're doing (its the vertical length from the chord to the surface of the sphere
the chord length is the key

Answer 19

Does the chord split into the middle of h?

Answer 20

using pythag: h^2 = a^2 + b^2
=>
r^2 = 0.5 ^2 + x ^2
2 = 0.25 + x^2
x= sqrt 1.75 ~~approx 1.323m

Answer 21

Your answer is correct but could you please explain it more to me

Answer 22

yep, no worries

Answer 23

Particularly the chord

Answer 24

Thanks jack

Answer 25

so we know that the cone has an opening with a radius less than that of the radius of the sphere.
If not, instead of sitting on top of the the sphere like a party hat, the sphere would be engulfed by the cone until it reaches a point where the 2 circumference are equal
ie a party hat on a shot put vs a party hat on a marble, the marble would roll up the hat till it hit a tight fit

Answer 26

so all that means is that the the side length of the cone is irrelevant for our purposes

Answer 27

Could you please draw it

Answer 28

next, it sits on top of the sphere, so as a sphere is a uniform object

Answer 29

sure

Answer 30

Oh now I see logically the radius of the sphere is larger than the radius of the cone

Answer 31

In the first graph I mean

Answer 32

for example 2, the side length matters as we would have to do further calculations to work out how high up the cone it sits and how much further down the base of the cone is so therefore how much larger the hole is

Answer 33

yep, u got it

Answer 34

Mmmmmm

Answer 35

I'm starting to get it now

Answer 36

so its a uniform object, so we can mentally bisect the sphere where it joins the cone and work with that
we're given that the base sits 0.5 m the center (one side of a triangle)
the hypotenuse we get from the radius (the distance from the edge of any line bisecting a sphere from the origin/certer is the radius)

Answer 37

So when the inverted cone is sitting on the sphere the base of it is above the centre of the sphere by 0.5 m. How could you put that on the graph

Answer 38

so we split the chord line (where we bisected) knowing that

see the 2nd image i drew:
its not to scale but h=0.5

Answer 39

Would you know of a graph that I can find off the web

Answer 40

hang on, ill google mathhelper

Answer 41

Your the best mate

Answer 42

http://mathhelpforum.com/geometry/184571-finding-diameter-slice-sphere.html
this is pretty much bang on, and explains better than I do
check out the pic down the bottom of the page

Answer 43

I drew it on a piece of paper

Answer 44

So basically the pointy side of the cone is standing in the centre of the sphere?

Answer 45

no, not at all man, the base of the cone is on top of the sphere like a hat, and we know how high up on the sphere it is resting (0.5m), then we use geometry and triangulation to work out that to be sitting at THAT height exactly, the circumference of the base of a cone is:
x

Answer 46

so all those lines are just triangulation, not the cone itself

Answer 47

Can I find a video tutorial on this

Answer 48

if the cone had a base of 2 x pi x sqrt 2, it would sit on the sphere covering exactly half of it (circumference = 2 pi r)

Answer 49

and from circumference we work our radius and vice versa
in ours we just worked out the radius of the base of the cone using triangulation

Answer 50

and not sure about videos, will aska a mod, standby
@amistre64 @Mertsj