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C
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Question Stats:
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48%
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What is the volume of a certain rectangular solid?
(1) Two adjacent faces of the solid have areas 15 and 24, respectively.
(2) Each of two opposite faces of the solid has area 40.
(1) Two adjacent faces of the solid have areas 15 and 24, respectively.
(2) Each of two opposite faces of the solid has area 40.
31 Aug 2010, 06:26
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This question is from the Official Guide and the Official Answer is C.
About the rectangular solid:
In a rectangular solid, all angles are right angles, and opposite faces are equal, so a rectangular solid can have a maximum of 3 different areas for its faces; on the diagram: yellow, green, and red faces can have different areas. I say at max because, for example, a rectangular solid can be a cube and in this case, it'll have all faces equal. Also, it's possible to have only 2 different areas for the faces, for example, when the base is square and the height does not equal the side of this square.
The volume of a rectangular solid is Volume = Length * Height * Depth.
BACK TO THE ORIGINAL QUESTION:
What is the volume of a certain rectangular solid?
(1) Two adjacent faces of the solid have areas 15 and 24, respectively.
Let the two adjacent faces be the blue and yellow faces on the diagram --> \(blue = dh = 15\) and \(yellow = lh = 24\) --> we have 2 equations with 3 unknowns; not sufficient to calculate the value of each or the product of the unknowns (\(V = lhd\)).
To elaborate more:
If \(blue = dh = 151 = 15\) and \(yellow = lh = 241 = 24\) then \(V = lhd = 24115 = 360\);
If \(blue = dh = 53 = 15\) and \(yellow = lh = 83 = 24\) then \(V = lhd = 835 = 120\).
Two different answer, hence not sufficient.
(2) Each of two opposite faces of the solid has area 40.
The above just gives the areas of two opposite faces, so clearly insufficient.
(1)+(2) From (1): \(blue = dh = 15\), \(yellow = lh = 24\), and from (2) each of two opposite faces of the solid has an area of 40, so it must be the red one: \(red = dl = 40\) --> here we have 3 distinct linear equations with 3 unknowns; hence we can find the values of each and thus can calculate \(V = lhd\). Sufficient.
To show how it can be done: multiply these 3 equations --> \(l^2h^2d^2 = (lhd)^2 = 15*24*40 = 24^25^2\) --> \(V = lhd = 24*5 = 120\).
Answer: C.
Hope it helps.
800px-Cuboid.png [ 22.06 KiB | Viewed 91358 times ]
About the rectangular solid:
In a rectangular solid, all angles are right angles, and opposite faces are equal, so a rectangular solid can have a maximum of 3 different areas for its faces; on the diagram: yellow, green, and red faces can have different areas. I say at max because, for example, a rectangular solid can be a cube and in this case, it'll have all faces equal. Also, it's possible to have only 2 different areas for the faces, for example, when the base is square and the height does not equal the side of this square.
The volume of a rectangular solid is Volume = Length * Height * Depth.
BACK TO THE ORIGINAL QUESTION:
What is the volume of a certain rectangular solid?
(1) Two adjacent faces of the solid have areas 15 and 24, respectively.
Let the two adjacent faces be the blue and yellow faces on the diagram --> \(blue = dh = 15\) and \(yellow = lh = 24\) --> we have 2 equations with 3 unknowns; not sufficient to calculate the value of each or the product of the unknowns (\(V = lhd\)).
To elaborate more:
If \(blue = dh = 151 = 15\) and \(yellow = lh = 241 = 24\) then \(V = lhd = 24115 = 360\);
If \(blue = dh = 53 = 15\) and \(yellow = lh = 83 = 24\) then \(V = lhd = 835 = 120\).
Two different answer, hence not sufficient.
(2) Each of two opposite faces of the solid has area 40.
The above just gives the areas of two opposite faces, so clearly insufficient.
(1)+(2) From (1): \(blue = dh = 15\), \(yellow = lh = 24\), and from (2) each of two opposite faces of the solid has an area of 40, so it must be the red one: \(red = dl = 40\) --> here we have 3 distinct linear equations with 3 unknowns; hence we can find the values of each and thus can calculate \(V = lhd\). Sufficient.
To show how it can be done: multiply these 3 equations --> \(l^2h^2d^2 = (lhd)^2 = 15*24*40 = 24^25^2\) --> \(V = lhd = 24*5 = 120\).
Answer: C.
Hope it helps.
Attachment:
800px-Cuboid.png [ 22.06 KiB | Viewed 91358 times ]
24 Feb 2010, 08:27
Kudos
Bookmarks
1) Two adjacent faces of the slid have areas 15 & 24, respectively.
2) Each of the opposite faces of the solid has area 40.
Let the length of the sides be a,b,c. Areas of each of the sides will be ab,bc,ca and the volume is abc
1. ab=15 ac=24 From these we cannot get individual values of a,b,c or the volume. So INSUFFICIENT
2. bc=40. From this we cannot get the volume. -- INSUFFICIENT
Combining we have:
ab=15
ac=24
bc=40
multiplying the LHS and RHS, you get:
(ab)(ac)(bc)=15*24*40
(a^2)(b^2)(c^2)=14400
or abc=120=Volume
So, C is the answer
2) Each of the opposite faces of the solid has area 40.
Let the length of the sides be a,b,c. Areas of each of the sides will be ab,bc,ca and the volume is abc
1. ab=15 ac=24 From these we cannot get individual values of a,b,c or the volume. So INSUFFICIENT
2. bc=40. From this we cannot get the volume. -- INSUFFICIENT
Combining we have:
ab=15
ac=24
bc=40
multiplying the LHS and RHS, you get:
(ab)(ac)(bc)=15*24*40
(a^2)(b^2)(c^2)=14400
or abc=120=Volume
So, C is the answer
