All Answers

+118725

Ok Chris, you were thinking the same as me
but why do you say

Quote:
the problem only "works" if we assume that AB is perpendicular to BC.

If AB is not perpendicular to BC don't we just need more information? That puts us straight back to the answer you and I gave in the first place. - or am I missing something?

MelodyApr 10, 2014

+33666

CPhill:
kayla121497 wrote:point A is the center of the sphere. point C is on the surface of the sphere. point B is the center of the circle that lies in plane P and includes point C. the radius of the circle is 12mm. AB= 5 mm. What is the volume of the sphere to the nearest cubic mm?

Mmmmm....I don't know about this.....here's a cross-section of a sphere where A is the center, C lies on the sphere's surface and AB =5, BC = 12 and BC lies in Plane "P"
(I think all of these meet the requirements of the problem).

Circle.JPG

Note that the radius of the sphere is ≈ 10 (At least as close to 10 as I could make it!!)

Maybe I'm reading too much into this problem, but I claim that we don't actually know that BC is at a right angle to AB

I guess my point is that we could have a right triangle, but as you see, it might not have to be. Note that ∠BAC is greater than a right angle

In hindsight, the problem only "works" if we assume that AB is perpendicular to BC.......and since we're not given any other information, then DavidQD and Alan would certainly be correct !!!

You need to complete the circle centred on B. You can see from the picture below (drawn to scale in Geogebra) that a circle based on BC as radius (as constructed by CPhill above) would not be wholly on the sphere. (BE is the extension of BC such that CE is the diameter of the circle)
sphere2.PNG

AlanApr 10, 2014

+130561

kayla121497 wrote:point A is the center of the sphere. point C is on the surface of the sphere. point B is the center of the circle that lies in plane P and includes point C. the radius of the circle is 12mm. AB= 5 mm. What is the volume of the sphere to the nearest cubic mm?

Mmmmm....I don't know about this.....here's a cross-section of a sphere where A is the center, C lies on the sphere's surface and AB =5, BC = 12 and BC lies in Plane "P"
(I think all of these meet the requirements of the problem).

Circle.JPG

Note that the radius of the sphere is ≈ 10 (At least as close to 10 as I could make it!!)

Maybe I'm reading too much into this problem, but I claim that we don't actually know that BC is at a right angle to AB

I guess my point is that we could have a right triangle, but as you see, it might not have to be. Note that ∠BAC is greater than a right angle

In hindsight, the problem only "works" if we assume that AB is perpendicular to BC.......and since we're not given any other information, then DavidQD and Alan would certainly be correct !!!

CPhillApr 10, 2014

+33666

kayla121497:
point A is the center of the sphere. point C is on the surface of the sphere. point B is the center of the circle that lies in plane P and includes point C. the radius of the circle is 12mm. AB= 5 mm. What is the volume of the sphere to the nearest cubic mm?

AB (5mm) and BC (12mm) are two sides of a right-angled triangle of which the radius of the sphere is the hypotenuse. The triangle is the well -known 5, 12, 13 triangle, so the radius of the sphere is 13mm. The volume follows from V = (4/3)*pi*r ³ as in DavidQD's reply above.

See the picture below.
Sphere.PNG

AlanApr 10, 2014

+118725

Hi DavidQD
I thought of this but I wasn't convinced that it would always be true. I'll have to think some more.

MelodyApr 10, 2014

+330

kayla121497:
point A is the center of the sphere. point C is on the surface of the sphere. point B is the center of the circle that lies in plane P and includes point C. the radius of the circle is 12mm. AB= 5 mm. What is the volume of the sphere to the nearest cubic mm?

An intuitive and quick method to solve: Treat the circle on the plane as a spherical segment (analogous to a chord) situated 5 mm below the center of the sphere.

Find the angle at common point C by the arc-Tan of (5mm/12mm). This returns an angle of 22.62 degrees.
Divide 5mm by the Sine of this angle (5mm)/Sin(22.62) =(5mm/ 0.384)=13mm. This value is the radius of the sphere.

Then use the sphere volume formula (4/3)Pi (r ³). (4/3)Pi (13 ³])=

(4/3)*pi* (13^3)

[size=150] The volume of the sphere= 9203 mm³[/size]

~~D~~

DavidQDApr 10, 2014

+130561

I think this website explains it better than I can.......He gives lots of examples with several types of functions - some easy, some more difficult and some that are "out there." (Take heart, you probably won't be seeing the "out there" ones very often )

http://tutorial.math.lamar.edu/Classes/CalcI/ChainRule.aspx

I hope this helps........

CPhillApr 10, 2014

#39

I helps to explain it with visuals

1st Pick Reveal Switch?
Nothing Box 1 Yes-Million
Nothing Box 1 No-Nothing
Nothing Box 2 Yes-Million
Nothing Box 2 No-Nothing
Million Box 1-or-2 Yes-Nothing
Million Box 1-or-2 No-Million
As you can see NOT switching ONLY benefits you when you first chose the million (1/3 Chance) and Switching benefits you when ever you choose nothing (2/3). It now comes down to: What is more likely 1/3 or 2/3?

GuestApr 10, 2014

+130561

Perhaps you could supply a picture??? It's difficult to answer without knowing exactly where B lies..

CPhillApr 10, 2014

3 apples 3
add 3 apples + 3
_______________ _____
equals 6 apples = 6

GuestApr 10, 2014

+118725

kayla121497:
point A is the center of the sphere. point C is on the surface of the sphere. point B is the center of the circle that lies in plane P and includes point C. the radius of the circle is 12mm. AB= 5 mm. What is the volume of the sphere to the nearest cubic mm?

I am wondering if there is something missing - like an angle maybe?

MelodyApr 10, 2014

The world has many difficult questions, but this is easily answered.

In our world,we see many illusions, and "the banana" is one of them. Our mind can trick us into seeing either what the heart truly wants, or what mind fears the most. For example, I see and hear magoes all the time, but can never fully enjoy their taste, because they are fake.

Bottom line: BANANAS AND MANGOES ARE FAKE! , but Batman is not.

GuestApr 10, 2014

+118725

username:
i dont understand equations of circles
this is a statement not a question

Why not - they are not that hard?

MelodyApr 10, 2014

+130561

2/3*x=16 what is x?

Hint....multiply both sides by the reciprocal of 2/3.........see if that will solve it for you........

CPhillApr 10, 2014

Apr 9, 2014

+130561

12 x
11 9
what is the answer to this proportion?

So we have
12 / 11 = x / 9
Cross-multiplication is the way to solve these problems...This gives us.....

12 * 9 = 11 * x ...thus.....
108 = 11 x........Now divide by 11 on both sides
108 / 11 = x
So, x ≈ 9.818

(Note, that we could have set up the proportion as (12 / x = 11 / 9).....the result would have been the same !!! (Check this for yourself.)

CPhillApr 9, 2014

+130561

6x^3 - 21x^2 + 9x

The first rule in factoring is to take out the greatest common factor, i.e., 3......so we have

3(2x^3 - 7x +3)......now forget the 3, we'll add it back later

So. let's see if we can factor (2x^3 - 7x +3)
We need to find something that multiplies to 2 (the first coefficient) and something else that multiplies to 3 (the last coefficient). The final goal is that the sum of these two products should equal the middle coefficient, (-7). Trial and error is the way to go here.
Let's try.... (2x + 1) (x + 3)....Nope.....I get a (+ 7x) and I need a (-7x)
Mmmmm...we're close !!! Since I know that (-3) * (-1) also gives me a (+3), let's try that.
(2x - 1) (x - 3)......Hey, this gives us what we need !!!

Now, go back and "stick" the "3" on the front.....So we have
3(2x - 1) (x - 3)

And that's it !!

CPhillApr 9, 2014

+130561

86 gallons / 6 beds = 14 1/3 gallons per bed

CPhillApr 9, 2014

+33666

Perhaps it's worth noting that the calculator on this site has an nCr button for calculating the number of ways of choosing r objects out of n. For example ₈C ₅ is done as nCr(8,5) giving:

nCr(8,5)

AlanApr 9, 2014

+118725

reinout-g,
I figure you will see this note.
Could you please put some straight forward probability questions like this on in your probability posts sometimes?
I need more practice.
That would be great,
Thank you
Melody.

MelodyApr 9, 2014

+118725

Alan's done all the hard work. I would just phrase it a little differently. I have to think really hard about these questions.

Question: A group of 8 friends (5 girls and 3 boys) plan to watch a movie, but they have only 5 tickets.

The number of ways you can choose 5 distinct people from 8 is 8C5 which you can get from your calculator but it is equal to 8!/(3!*5!) = 56

The number of ways you can choose 3 boys from 3 boys is 1.
The number of ways you can choose 2 girls from 5 is 5C2
So the number of ways you can choose 3 boys and 2 girls is 1 * 5C2 = 5C2 = 10

So the number of ways of NOT choosing 3 boys and 2 girls must be 56-10 = 46
So the probabiliy of NOT choosing 3 boys and 2 girls is the same as the probablity of choosing 3 or more girls = 46/56

This is often presented slightly differently.
The P( 3 boys and 2 girls) is 10/56
so P( NOT choosing 3 boys and 2 girls) = 1- (10/56) = 46/56

I hope that also helped.

MelodyApr 9, 2014

+33666

unknow:
A group of 8 friends (5 girls and 3 boys) plans to watch a movie, but they have only 5 tickets. If they randomly decide who will watch the movie, what is the probability that there are at least 3 girls in the group that watch the movie?

First, the number of ways you can choose 5 people from 8 is given by 8!/(5!*3!) which is

8!/(5!*3!)

There will be at least three girls except in the situation when all three boys are chosen. The number of ways can you choose 3 boys from 5 people is 5!/(3!*2!) which is

5!/(3!*2!)

So the probability that there are at least 3 girls is (56 - 10)/56 or 46/56 = 0.821

I've assumed you know that the exclamation mark stands for "factorial". That is n! = n*(n-1)*(n-2)*....*2*1

Just in case you are uncertain about the above formulae, I'll illustrate all the possible ways of choosing 3 boys and 2 girls below.
Each column below represents the tickets in the order in which they are drawn.

b b b g g ... 1
b b g b g ... 2
b b g g b ... 3
b g b b g ... 4
b g b g b ... 5
b g g b b ... 6
g b b b g ... 7
g b b g b ... 8
g b g b b ... 9
g g b b b ... 10

AlanApr 9, 2014

#96

+118725

Hi all,
As always there were some great questions and answers posted today.
The answers were delivered by from Alan, CPhill, unknown, Zahcarry123, Rom, Michael Fennema and reinout-g. Thank you every one.
That is it for this Monday/Tuesday. Enjoy the rest of your day.
Melody.

MelodyApr 9, 2014

#38

+118725

I don't know reinout,

If p =0 then I won't confess and we both get 3 years
If p = 1 then I do confess and we both get 10 years.
If p is something else then I think I would have to leave it to the gods because I DO NOT KNOW

MelodyApr 9, 2014

#37

+2354

Melody:
Quote:
Hard one :

You and your best friend have been arrested for several white collar crimes. (Or something else which is more classy than stealing a candy bar)
Your attorney has told you the police has only got enough evidence to put you and your friend behind bars for 3 years.
However, the both of you are given the option to confess. If both of you confess you will both serve 10 years,
if one of you confesses and the other does not, the one who did not confess will serve 20 years.
Obviously, if you both remain silent you will both serve 3 years.
You are not allowed to have any communication with your friend.
Suppose your main priority is to keep yourself out of jail, and your secondary priority is to keep your partner out of jail.
You have decided that you find it twice as important to keep yourself out of jail as you find it important to keep your partner out of jail.
(So for example, in your opinion him serving 10 years is equal to you serving 5).
You estimate the chance of your partner confessing p.
At what value of p will you choose to confess and at what value will you keep silent?

Maybe it doesn't make any difference to the question BUT If one confesses and the other doesn't then the non confessor gets 20yrs. HOW many years will the confessor get?

The confessor will get 0 years, but remember you do care about your friend. You want to keep yourself out of jail twice as much as you want to keep your partner out of jail. Therefore, you'd rather serve 4 years than have your friend serve 10 years (since 2*4 = 8 < 10), but you'd be indifferent between you serving 4 years and him serving 8. (Yes this puzzle does require some calculation)

edit: I just made an edit to the original question, I changed 'at which value of p' to 'for which values of p'. Sorry if I might have put you on the wrong foot

reinout-gApr 9, 2014

#36

+118725

Quote:
Hard one :

You and your best friend have been arrested for several white collar crimes. (Or something else which is more classy than stealing a candy bar)
Your attorney has told you the police has only got enough evidence to put you and your friend behind bars for 3 years.
However, the both of you are given the option to confess. If both of you confess you will both serve 10 years,
if one of you confesses and the other does not, the one who did not confess will serve 20 years.
Obviously, if you both remain silent you will both serve 3 years.
You are not allowed to have any communication with your friend.
Suppose your main priority is to keep yourself out of jail, and your secondary priority is to keep your partner out of jail.
You have decided that you find it twice as important to keep yourself out of jail as you find it important to keep your partner out of jail.
(So for example, in your opinion him serving 10 years is equal to you serving 5).
You estimate the chance of your partner confessing p.
At what value of p will you choose to confess and at what value will you keep silent?

Maybe it doesn't make any difference to the question BUT If one confesses and the other doesn't then the non confessor gets 20yrs. HOW many years will the confessor get?

MelodyApr 9, 2014

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