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Direction: A cuboid is divided into 192 identical cubelets This is done by making minimum no. ot cuts possible. All cuts are parallel to some of the face. But before doing so. The cube is painted with green color on one set of opposite faces. Blue on other set of opposite faces and red on their pair of annosit faces.
What is the number of cubelets possible which are painted with none of the color?
- a)16
- b)24
- c)36
- d)48
Correct answer is option 'D'. Can you explain this answer?
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Direction: A cuboid is divided into 192 identical cubelets This is do...
Number of unpainted cubelets will be (4 -2) x (6 - 2) x (8 - 2) = 2 x 4 x 6 = 48
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Hence option (d)
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Direction: A cuboid is divided into 192 identical cubelets This is do...
Solution:
Given, a cuboid is divided into 192 identical cubelets with minimum no. of cuts possible. All cuts are parallel to some of the face. And, the cube is painted with green color on one set of opposite faces, blue on other set of opposite faces and red on their pair of annosit faces. We need to find the number of cubelets possible which are painted with none of the color.
To solve this problem, we need to follow the steps given below:
Step 1: Find the dimensions of the cuboid
Let the length, breadth and height of the cuboid be l, b and h respectively.
Given, the cuboid is divided into 192 identical cubelets.
Therefore, the volume of the cuboid = 192 x volume of a cubelet
=> l x b x h = 192 x (side)^3
=> (l x b x h)/192 = (side)^3 …(1)
Step 2: Find the length of the side of the cubelet
The cubelet is formed by dividing the cuboid into smaller cubes with minimum no. of cuts possible. Hence, the length of the side of the cubelet must be a factor of l, b and h.
Also, from equation (1), we know that (side)^3 = (l x b x h)/192.
Therefore, we need to find the common factors of l, b and h which divide (l x b x h)/192.
Let us write l, b and h in terms of their prime factors as follows:
l = 2^a x 3^b x 5^c x ....
b = 2^p x 3^q x 5^r x ....
h = 2^x x 3^y x 5^z x ....
where a, b, c, p, q, r, x, y, z, ... are non-negative integers.
Then, (l x b x h)/192 = 2^(a+p+x-6) x 3^(b+q+y-1) x 5^(c+r+z-1) x ....
Now, we need to find the common factors of l, b and h which divide (l x b x h)/192. This is possible only if the exponents of 2, 3 and 5 in (l x b x h)/192 are at least 6, 1 and 1 respectively.
Hence, we can choose a, p and x such that a+p+x = 6 or 7 or 8 or 9 or 10 or 11.
Similarly, we can choose b, q and y such that b+q+y = 1 or 2.
And, we can choose c, r and z such that c+r+z = 1 or 2.
Step 3: Find the number of cubelets which are painted with none of the colors
There are 3 sets of opposite faces in the cuboid which are painted with green, blue and red colors respectively. Hence, each cubelet must have at least one face which is painted with a color.
To find the number of cubelets which are painted with none of the colors, we need to find the number of cubelets which have all their faces unpainted.
Such cubelets can be formed by choosing any one of the a, p,
Given, a cuboid is divided into 192 identical cubelets with minimum no. of cuts possible. All cuts are parallel to some of the face. And, the cube is painted with green color on one set of opposite faces, blue on other set of opposite faces and red on their pair of annosit faces. We need to find the number of cubelets possible which are painted with none of the color.
To solve this problem, we need to follow the steps given below:
Step 1: Find the dimensions of the cuboid
Let the length, breadth and height of the cuboid be l, b and h respectively.
Given, the cuboid is divided into 192 identical cubelets.
Therefore, the volume of the cuboid = 192 x volume of a cubelet
=> l x b x h = 192 x (side)^3
=> (l x b x h)/192 = (side)^3 …(1)
Step 2: Find the length of the side of the cubelet
The cubelet is formed by dividing the cuboid into smaller cubes with minimum no. of cuts possible. Hence, the length of the side of the cubelet must be a factor of l, b and h.
Also, from equation (1), we know that (side)^3 = (l x b x h)/192.
Therefore, we need to find the common factors of l, b and h which divide (l x b x h)/192.
Let us write l, b and h in terms of their prime factors as follows:
l = 2^a x 3^b x 5^c x ....
b = 2^p x 3^q x 5^r x ....
h = 2^x x 3^y x 5^z x ....
where a, b, c, p, q, r, x, y, z, ... are non-negative integers.
Then, (l x b x h)/192 = 2^(a+p+x-6) x 3^(b+q+y-1) x 5^(c+r+z-1) x ....
Now, we need to find the common factors of l, b and h which divide (l x b x h)/192. This is possible only if the exponents of 2, 3 and 5 in (l x b x h)/192 are at least 6, 1 and 1 respectively.
Hence, we can choose a, p and x such that a+p+x = 6 or 7 or 8 or 9 or 10 or 11.
Similarly, we can choose b, q and y such that b+q+y = 1 or 2.
And, we can choose c, r and z such that c+r+z = 1 or 2.
Step 3: Find the number of cubelets which are painted with none of the colors
There are 3 sets of opposite faces in the cuboid which are painted with green, blue and red colors respectively. Hence, each cubelet must have at least one face which is painted with a color.
To find the number of cubelets which are painted with none of the colors, we need to find the number of cubelets which have all their faces unpainted.
Such cubelets can be formed by choosing any one of the a, p,
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Direction: A cuboid is divided into 192 identical cubelets This is do...
D
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Direction: A cuboid is divided into 192 identical cubelets This is done by making minimum no. ot cuts possible. All cuts are parallel to some of the face. But before doing so. The cube is painted with green color on one set of opposite faces. Blue on other set of opposite faces and red on their pair of annosit faces.What is the number of cubelets possible which are painted with none of the color?a)16b)24c)36d)48Correct answer is option 'D'. Can you explain this answer?
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Direction: A cuboid is divided into 192 identical cubelets This is done by making minimum no. ot cuts possible. All cuts are parallel to some of the face. But before doing so. The cube is painted with green color on one set of opposite faces. Blue on other set of opposite faces and red on their pair of annosit faces.What is the number of cubelets possible which are painted with none of the color?a)16b)24c)36d)48Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Direction: A cuboid is divided into 192 identical cubelets This is done by making minimum no. ot cuts possible. All cuts are parallel to some of the face. But before doing so. The cube is painted with green color on one set of opposite faces. Blue on other set of opposite faces and red on their pair of annosit faces.What is the number of cubelets possible which are painted with none of the color?a)16b)24c)36d)48Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Direction: A cuboid is divided into 192 identical cubelets This is done by making minimum no. ot cuts possible. All cuts are parallel to some of the face. But before doing so. The cube is painted with green color on one set of opposite faces. Blue on other set of opposite faces and red on their pair of annosit faces.What is the number of cubelets possible which are painted with none of the color?a)16b)24c)36d)48Correct answer is option 'D'. Can you explain this answer?.
Direction: A cuboid is divided into 192 identical cubelets This is done by making minimum no. ot cuts possible. All cuts are parallel to some of the face. But before doing so. The cube is painted with green color on one set of opposite faces. Blue on other set of opposite faces and red on their pair of annosit faces.What is the number of cubelets possible which are painted with none of the color?a)16b)24c)36d)48Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Direction: A cuboid is divided into 192 identical cubelets This is done by making minimum no. ot cuts possible. All cuts are parallel to some of the face. But before doing so. The cube is painted with green color on one set of opposite faces. Blue on other set of opposite faces and red on their pair of annosit faces.What is the number of cubelets possible which are painted with none of the color?a)16b)24c)36d)48Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Direction: A cuboid is divided into 192 identical cubelets This is done by making minimum no. ot cuts possible. All cuts are parallel to some of the face. But before doing so. The cube is painted with green color on one set of opposite faces. Blue on other set of opposite faces and red on their pair of annosit faces.What is the number of cubelets possible which are painted with none of the color?a)16b)24c)36d)48Correct answer is option 'D'. Can you explain this answer?.
Solutions for Direction: A cuboid is divided into 192 identical cubelets This is done by making minimum no. ot cuts possible. All cuts are parallel to some of the face. But before doing so. The cube is painted with green color on one set of opposite faces. Blue on other set of opposite faces and red on their pair of annosit faces.What is the number of cubelets possible which are painted with none of the color?a)16b)24c)36d)48Correct answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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Direction: A cuboid is divided into 192 identical cubelets This is done by making minimum no. ot cuts possible. All cuts are parallel to some of the face. But before doing so. The cube is painted with green color on one set of opposite faces. Blue on other set of opposite faces and red on their pair of annosit faces.What is the number of cubelets possible which are painted with none of the color?a)16b)24c)36d)48Correct answer is option 'D'. Can you explain this answer?, a detailed solution for Direction: A cuboid is divided into 192 identical cubelets This is done by making minimum no. ot cuts possible. All cuts are parallel to some of the face. But before doing so. The cube is painted with green color on one set of opposite faces. Blue on other set of opposite faces and red on their pair of annosit faces.What is the number of cubelets possible which are painted with none of the color?a)16b)24c)36d)48Correct answer is option 'D'. Can you explain this answer? has been provided alongside types of Direction: A cuboid is divided into 192 identical cubelets This is done by making minimum no. ot cuts possible. All cuts are parallel to some of the face. But before doing so. The cube is painted with green color on one set of opposite faces. Blue on other set of opposite faces and red on their pair of annosit faces.What is the number of cubelets possible which are painted with none of the color?a)16b)24c)36d)48Correct answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
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