A figure shows a cubical box with a sphere that just fits inside.That is,the length,width,and height of the box are each equal to the diameter of the ball.What percent to the nearest tenth of the volume of the box is not occupied by the ball?

- anonymous

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- katieb

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- amistre64

lol again?

- amistre64

x^3 - still dont know the formula for a Sphere volume = free volume

- amistre64

4pir^3
------ ?? sounds familiar
3

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- anonymous

I have 4/3 pi r^3 as the sphere volume

- anonymous

I don't understand sphere volume = free volume

- amistre64

r = x so lets run with that.
it means that the sphere eats up so much of the boxes volume that whatever is not taken up by the sphere is "free"....empty space

- amistre64

x^3 - 4pi(x^3)
-------
3
3x^3 - 4pi(x^3)
--------------
3
x^3(3 - 4pi)
---------- should be the free space in the box.
3

- anonymous

What is the box volume to minus the sphere volume?

- amistre64

suppose you have a bathtub filled with bubble bath till lits almost overflowing.... when you get in, the volume of water that flows out is taken up by the volume of your body getting in; when you get out of the tub, the amount of water left in is the stuff that didnt overflow to begin with and it is equal to:
volume of tub - volume of you...

- anonymous

o.k. I can imagine that,but in order for me to resolve this problem I need the formula of the volume for the box and sphere.

- amistre64

that is correct:
the volume of a square cubical box is:
side times side times side...or side^3; lets call it x^3 :)

- amistre64

the radius of the sphere is x/2 so my initial figureing was misappropriated.....

- anonymous

am I suppose to subtract something?

- anonymous

how do I input the information into my calculator?

- anonymous

What is the percent?

- amistre64

do you have a "real" value for the sides of the box? or the radius of the sphere?
without that, you are left with a formula with variables in it

- amistre64

if we assume the box side is twice the radius of the sphere; we get:
4pi(x^3)
8x^3 - ------- for the free volume in the box
3

- anonymous

The problem is how it's stated -no numbers.

- anonymous

the length,width,and height of the box are equal to the diameter of the ball

- amistre64

24x^3 - 4pi(x^3)
---------------- x100 = percent free
24x^3

- anonymous

I don't understand how you got your numbers

- amistre64

the volume of a box is (8x^3)
2x * 2x * 2x = 8x^3

- anonymous

I need the percent to the nearest tenth of a volume of the box not occupied by the ball

- amistre64

do you agree that the volume of the box is 8x^3?
if there is a figure that goes with the question....tell me, does it show a number for the raduis of the sphere or perhaps a number for the side of the box?

- anonymous

Nope,the illustration only displays a ball inside of a cubicle box

- amistre64

then we will have to see if our variables cancel out at some point in the process........ when we get to the end of the process, lets see if we get a "percentage" that doesnt include a variable then.... you agree?

- anonymous

Yes,agree.

- amistre64

I beleive I was up to here in the process then :)
24x^3 - 4pi(x^3)
---------------- x100 = percent free
24x^3

- anonymous

I'm just having a difficult time with the process.

- amistre64

I know.... but what we are doing is just substituting letter for number and working it thru.....

- amistre64

lets say the radius of the sphere is equal to "x"
the radius of the sphere is exactly "half" the measurement of a side of the box;
so 2x = the width, the height and the depth of the box.
w*h*d = 2x * 2x * 2x = 8x^3 you agree?

- anonymous

So,if I were to input your equation into my calculator I got the answer: 13823.9091

- amistre64

i havent checked my own propensity for error yet :) lets get to the end of the process first ok?

- amistre64

volume of box = 8x^3
volume of sphere = 4pi(x^3)
-------- right?
3

- anonymous

o.k. then what?

- amistre64

(3)8x^3 4pi(x^3) 24(x^3) - 4pi(x^3)
------- - ------- = ----------------
3 3 3

- amistre64

which equals:
4x^3(6 - pi)
----------
3
we divide this "value" by the volume of the box: 8x^3
4x^3(6 - pi)
---------- / 8x^3
3
4x^3(6 - pi)
---------- we can cancel out like terms and reduce this.
24x^3

- anonymous

is this the simplest,straightforward method?

- amistre64

(6-pi)
----- = 1 - pi/6 this is a decimal value
6
we multiply it by 100 to get a number for the %
100(1-pi/6) = 100 - 50pi/3. the calculator says
.........its the simplest and straight forward answer yes; do you really think Id waste my time helping you for a joke?.....dont interupt :)

- anonymous

:) sorry.

- amistre64

47.64 % is the result I get :)

- anonymous

Wow! So why do we have the equations over 3?

- amistre64

becasue 100pi/6 reduces to 50pi/3

- amistre64

we could stop at:
100(1-(pi/6)) and get the same results i spose :)

- anonymous

Hmm...I really need to look over the whole thing and digest this.

- amistre64

you do that..... and since the "x" variable vanishes in the end, we probably could have went with a value of x=1 to begin with; but i wasnt sure if it was going to work out that well :)

- anonymous

O.k. well,thank you for your trouble and time.

- amistre64

youre welcome :)

- anonymous

oh.... amistre :)

- amistre64

....yes???

- anonymous

help please <3

- amistre64

....post a new question so I aint gotta scroll :)

- anonymous

ok

- anonymous

Write the trigonometric expression in terms of sine and cosine, and then simplify.
sec(x)/csc(x)

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