Mixing
All Points Distance of 2 Away from the Unit Cube in R3
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The official question reads:
"Find the volume of the region in R3 consisting of all points whose distance
to the unit cube {(x, y, z) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1} is less than or equal to 2."
We are supposed to use "basic solid shapes" to solve this problem. My intuition tells me that there would be an eighth of a sphere, each with radius 2, surrounding each of the 8 vertices, as well as rectangular prisms with side lengths of 2,1,1 aligned with each of the 6 faces. So my result was 1/8*8*(4/3pi*2^3) + 6*2*1*1 = 12 + 32/3 pi. Does my reasoning make sense? Am I missing any shapes?
calculus geometry
asked Jan 28 at 22:48
Farooq AnsariFarooq Ansari
112
$endgroup$
add a comment |
$begingroup$
The official question reads:
"Find the volume of the region in R3 consisting of all points whose distance
to the unit cube {(x, y, z) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1} is less than or equal to 2."
We are supposed to use "basic solid shapes" to solve this problem. My intuition tells me that there would be an eighth of a sphere, each with radius 2, surrounding each of the 8 vertices, as well as rectangular prisms with side lengths of 2,1,1 aligned with each of the 6 faces. So my result was 1/8*8*(4/3pi*2^3) + 6*2*1*1 = 12 + 32/3 pi. Does my reasoning make sense? Am I missing any shapes?
calculus geometry
asked Jan 28 at 22:48
Farooq AnsariFarooq Ansari
112
$endgroup$
2
$begingroup$
You are on the right track, but are not quite there yet. Have you considered what points are at distance $2$ from the edges?
$endgroup$
– Servaes
Jan 28 at 22:55
1
$begingroup$
You forget two pieces, 1) the unit cube itself. 2) the 12 quarter cylinders of length 1 and radius $2$ along the edges (as pointed out by Servaes before)
$endgroup$
– achille hui
Jan 28 at 23:13
$begingroup$
Welcome to MSE. Please edit and use MathJax to properly format math expressions.
$endgroup$
– Lee David Chung Lin
Jan 29 at 0:41
add a comment |
$begingroup$
The official question reads:
"Find the volume of the region in R3 consisting of all points whose distance
to the unit cube {(x, y, z) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1} is less than or equal to 2."
We are supposed to use "basic solid shapes" to solve this problem. My intuition tells me that there would be an eighth of a sphere, each with radius 2, surrounding each of the 8 vertices, as well as rectangular prisms with side lengths of 2,1,1 aligned with each of the 6 faces. So my result was 1/8*8*(4/3pi*2^3) + 6*2*1*1 = 12 + 32/3 pi. Does my reasoning make sense? Am I missing any shapes?
calculus geometry
asked Jan 28 at 22:48
Farooq AnsariFarooq Ansari
112
$endgroup$
The official question reads:
"Find the volume of the region in R3 consisting of all points whose distance
to the unit cube {(x, y, z) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1} is less than or equal to 2."
We are supposed to use "basic solid shapes" to solve this problem. My intuition tells me that there would be an eighth of a sphere, each with radius 2, surrounding each of the 8 vertices, as well as rectangular prisms with side lengths of 2,1,1 aligned with each of the 6 faces. So my result was 1/8*8*(4/3pi*2^3) + 6*2*1*1 = 12 + 32/3 pi. Does my reasoning make sense? Am I missing any shapes?
calculus geometry
calculus geometry
asked Jan 28 at 22:48
Farooq AnsariFarooq Ansari
112
asked Jan 28 at 22:48
Farooq AnsariFarooq Ansari
112
asked Jan 28 at 22:48
Farooq AnsariFarooq Ansari
112
asked Jan 28 at 22:48
Farooq AnsariFarooq Ansari
112
asked Jan 28 at 22:48
Farooq AnsariFarooq Ansari
112
112
2
$begingroup$
You are on the right track, but are not quite there yet. Have you considered what points are at distance $2$ from the edges?
$endgroup$
– Servaes
Jan 28 at 22:55
1
$begingroup$
You forget two pieces, 1) the unit cube itself. 2) the 12 quarter cylinders of length 1 and radius $2$ along the edges (as pointed out by Servaes before)
$endgroup$
– achille hui
Jan 28 at 23:13
$begingroup$
Welcome to MSE. Please edit and use MathJax to properly format math expressions.
$endgroup$
– Lee David Chung Lin
Jan 29 at 0:41
add a comment |
2
$begingroup$
You are on the right track, but are not quite there yet. Have you considered what points are at distance $2$ from the edges?
$endgroup$
– Servaes
Jan 28 at 22:55
1
$begingroup$
You forget two pieces, 1) the unit cube itself. 2) the 12 quarter cylinders of length 1 and radius $2$ along the edges (as pointed out by Servaes before)
$endgroup$
– achille hui
Jan 28 at 23:13
$begingroup$
Welcome to MSE. Please edit and use MathJax to properly format math expressions.
$endgroup$
– Lee David Chung Lin
Jan 29 at 0:41
2
2
$begingroup$
You are on the right track, but are not quite there yet. Have you considered what points are at distance $2$ from the edges?
$endgroup$
– Servaes
Jan 28 at 22:55
$begingroup$
You are on the right track, but are not quite there yet. Have you considered what points are at distance $2$ from the edges?
$endgroup$
– Servaes
Jan 28 at 22:55
1
1
$begingroup$
You forget two pieces, 1) the unit cube itself. 2) the 12 quarter cylinders of length 1 and radius $2$ along the edges (as pointed out by Servaes before)
$endgroup$
– achille hui
Jan 28 at 23:13
$begingroup$
You forget two pieces, 1) the unit cube itself. 2) the 12 quarter cylinders of length 1 and radius $2$ along the edges (as pointed out by Servaes before)
$endgroup$
– achille hui
Jan 28 at 23:13
$begingroup$
Welcome to MSE. Please edit and use MathJax to properly format math expressions.
$endgroup$
– Lee David Chung Lin
Jan 29 at 0:41
$begingroup$
Welcome to MSE. Please edit and use MathJax to properly format math expressions.
$endgroup$
– Lee David Chung Lin
Jan 29 at 0:41
add a comment |
0
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$begingroup$
You are on the right track, but are not quite there yet. Have you considered what points are at distance $2$ from the edges?
$endgroup$
– Servaes
Jan 28 at 22:55
1
$begingroup$
You forget two pieces, 1) the unit cube itself. 2) the 12 quarter cylinders of length 1 and radius $2$ along the edges (as pointed out by Servaes before)
$endgroup$
– achille hui
Jan 28 at 23:13
$begingroup$
Welcome to MSE. Please edit and use MathJax to properly format math expressions.
$endgroup$
– Lee David Chung Lin
Jan 29 at 0:41