Mixing
AIME I 2000 Problem 8: Calculating the height of the liquid given Fraction of the Volume
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As the title says, I'm looking at problem number 8 from AIME I 2000.
https://artofproblemsolving.com/wiki/index.php?title=2000_AIME_I_Problems/Problem_8
I'm currently looking at solution 3 and I understand everything up to the point where it says the cone that is filled with air must have a height of the cube root of the fraction of the cone that the air occupies.
What am I missing here? Where does one get cube root from anything?
contest-math 3d volume
edited Jul 28 at 11:35
Ninja hatori
105113
asked Jul 28 at 9:03
ShadyAF
288
add a comment |Â
up vote
0
down vote
favorite
As the title says, I'm looking at problem number 8 from AIME I 2000.
https://artofproblemsolving.com/wiki/index.php?title=2000_AIME_I_Problems/Problem_8
I'm currently looking at solution 3 and I understand everything up to the point where it says the cone that is filled with air must have a height of the cube root of the fraction of the cone that the air occupies.
What am I missing here? Where does one get cube root from anything?
contest-math 3d volume
edited Jul 28 at 11:35
Ninja hatori
105113
asked Jul 28 at 9:03
ShadyAF
288
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
As the title says, I'm looking at problem number 8 from AIME I 2000.
https://artofproblemsolving.com/wiki/index.php?title=2000_AIME_I_Problems/Problem_8
I'm currently looking at solution 3 and I understand everything up to the point where it says the cone that is filled with air must have a height of the cube root of the fraction of the cone that the air occupies.
What am I missing here? Where does one get cube root from anything?
contest-math 3d volume
edited Jul 28 at 11:35
Ninja hatori
105113
asked Jul 28 at 9:03
ShadyAF
288
As the title says, I'm looking at problem number 8 from AIME I 2000.
https://artofproblemsolving.com/wiki/index.php?title=2000_AIME_I_Problems/Problem_8
I'm currently looking at solution 3 and I understand everything up to the point where it says the cone that is filled with air must have a height of the cube root of the fraction of the cone that the air occupies.
What am I missing here? Where does one get cube root from anything?
contest-math 3d volume
edited Jul 28 at 11:35
Ninja hatori
105113
asked Jul 28 at 9:03
ShadyAF
288
edited Jul 28 at 11:35
Ninja hatori
105113
edited Jul 28 at 11:35
Ninja hatori
105113
edited Jul 28 at 11:35
Ninja hatori
105113
105113
asked Jul 28 at 9:03
ShadyAF
288
asked Jul 28 at 9:03
ShadyAF
288
asked Jul 28 at 9:03
ShadyAF
288
288
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
For a given right circular cone, the ratio of the radius $r$ to the height $h$ is constant. In this instance that ratio is $fracrh = frac512 = k$. The volume $V_1$ at a given height $h_1$ is thus $V_1 = fracpi (kh_1)^2 h_13=fracpi k^23 h_1^3$. The volume at another height $h_2$ is likewise $V_2=fracpi k^23 h_2^3$. If the ratio of volumes is known, i.e. $fracV_1V_2 = C$, the ratio of the heights will also be known, i.e. $h_1 = C^frac13h_2$.
answered Jul 28 at 11:38
Jens
3,0262826
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
For a given right circular cone, the ratio of the radius $r$ to the height $h$ is constant. In this instance that ratio is $fracrh = frac512 = k$. The volume $V_1$ at a given height $h_1$ is thus $V_1 = fracpi (kh_1)^2 h_13=fracpi k^23 h_1^3$. The volume at another height $h_2$ is likewise $V_2=fracpi k^23 h_2^3$. If the ratio of volumes is known, i.e. $fracV_1V_2 = C$, the ratio of the heights will also be known, i.e. $h_1 = C^frac13h_2$.
answered Jul 28 at 11:38
Jens
3,0262826
add a comment |Â
up vote
1
down vote
accepted
For a given right circular cone, the ratio of the radius $r$ to the height $h$ is constant. In this instance that ratio is $fracrh = frac512 = k$. The volume $V_1$ at a given height $h_1$ is thus $V_1 = fracpi (kh_1)^2 h_13=fracpi k^23 h_1^3$. The volume at another height $h_2$ is likewise $V_2=fracpi k^23 h_2^3$. If the ratio of volumes is known, i.e. $fracV_1V_2 = C$, the ratio of the heights will also be known, i.e. $h_1 = C^frac13h_2$.
answered Jul 28 at 11:38
Jens
3,0262826
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
For a given right circular cone, the ratio of the radius $r$ to the height $h$ is constant. In this instance that ratio is $fracrh = frac512 = k$. The volume $V_1$ at a given height $h_1$ is thus $V_1 = fracpi (kh_1)^2 h_13=fracpi k^23 h_1^3$. The volume at another height $h_2$ is likewise $V_2=fracpi k^23 h_2^3$. If the ratio of volumes is known, i.e. $fracV_1V_2 = C$, the ratio of the heights will also be known, i.e. $h_1 = C^frac13h_2$.
answered Jul 28 at 11:38
Jens
3,0262826
For a given right circular cone, the ratio of the radius $r$ to the height $h$ is constant. In this instance that ratio is $fracrh = frac512 = k$. The volume $V_1$ at a given height $h_1$ is thus $V_1 = fracpi (kh_1)^2 h_13=fracpi k^23 h_1^3$. The volume at another height $h_2$ is likewise $V_2=fracpi k^23 h_2^3$. If the ratio of volumes is known, i.e. $fracV_1V_2 = C$, the ratio of the heights will also be known, i.e. $h_1 = C^frac13h_2$.
answered Jul 28 at 11:38
Jens
3,0262826
answered Jul 28 at 11:38
Jens
3,0262826
answered Jul 28 at 11:38
Jens
3,0262826
answered Jul 28 at 11:38
Jens
3,0262826
3,0262826
add a comment |Â
add a comment |Â
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