A empty 1 km cube has a potent source of deadly radium at its center. The danger zone is anywhere within 3/4 km. What's the volume of the danger zone inside the cube?
http://answers.yahoo.com/question/index;_ylt=AteE6HasoAEfawqw2Dvk9avpy6IX;_ylv=3?qid=20071223114207AAi1r3l
2007-12-23 10:42:18 · 6 answers · asked by Scythian1950 7 in Science & Mathematics ➔ Mathematics
Volume, not volune.
2007-12-23 10:42:38 · update #1
John, a volune is a volume that has got lunes.
2007-12-23 10:47:40 · update #2
Within 3/4 km of the CENTER.
2007-12-23 10:54:37 · update #3
Psst, the answer is going to be slighly under 1 cubic km, since it's just the corners truncated.
2007-12-23 15:30:35 · update #4
Yes, adding the caps overestimates the volume, because they overlap. The answer is less than 5π/16.
2007-12-24 11:57:11 · update #5
Can't solve it now. Got my six-year olds playing football inside the house. But at least l can read (and without spell check, I invariably spell volume as "volumn.")
For those that follow, there is a portion of the cube which is outside of the danger zone. The corners of the cube are more than 3/4 of a km from the center. If someone doesn't solve before my kids bedtime, I'll give it a try.
>>>>>>>> Edit >>>>>>>
To late to attempt this. 10:00 pm and I'm tired.
But the cap method looks good -- except that all of THE CAPS OVERLAP. It is just those little corners. The cap method overestimates the volume.
2007-12-23 10:59:17 · answer #1 · answered by Frst Grade Rocks! Ω 7 · 0⤊ 0⤋
One sixth the volume of the portion of a sphere of radius 3/4 that would be outside the cube is the volume obtained by rotating the region bounded by x² + y² = (3/4)² and x = 1/2 about the x-axis:
â« {1/2 to 3/4} Ï (9/16 - x²) dx = Ï/24,
so the volume of the danger zone inside the cube is
(4/3)Ï(3/4)³ - 6(Ï/24) = 5Ï/16
2007-12-23 19:24:51 · answer #2 · answered by a²+b²=c² 4 · 1⤊ 0⤋
I take it 1 km is the length of the side of the cube, but 3/4 km is the radius of the sphere. Then the cube cuts 6 caps off the sphere.
Volume of the sphere is 4Ïr³/3 = 4Ï(3/4)³/3 = 9Ï/16.
Each cap will have a height of 1/4 km and a radius r such that r² = (3/4)² - (1/2)² = 5/16, so r = (â5)/4.
Volume of each cap will be (Ï/6)(3r² + h²)h = (Ï/6)(3•5/16 + 1/16)(1/4) = (Ï/6)(1)(1/4) = Ï/24. Then 6 caps have a volume of Ï/4.
That makes the volume of the shape 9Ï/16 - Ï/4 = 5Ï/16 km³.
2007-12-23 19:10:32 · answer #3 · answered by Philo 7 · 3⤊ 0⤋
The entire cube lies within .75 Km. of the center,
so the requested volume is 1 Km.^3
2007-12-23 19:32:12 · answer #4 · answered by Irv S 7 · 0⤊ 1⤋
the volume of danger is a sphere of radius 3/4km centered on the source of radiation
the volume of a sphere is V=4/3 pi R^3
so the volume is 4/3 pi (3/4)^3 = 1.76 km^3
2007-12-23 18:49:15 · answer #5 · answered by kuiperbelt2003 7 · 0⤊ 1⤋
scythian, what's a volune?---lol 1 x .75 x 4 = 3?
2007-12-23 18:46:28 · answer #6 · answered by "Johns" 7 · 1⤊ 0⤋