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Cubes of same size are arranged in the form of cube on a table then a column of five cubes is removed from each of the four cubes is removed from each of the four corners. All the exposed faces of rest of the cube are coloured yellow. Then how many small cubes are there in the solid after the removal of the column?
- a)100
- b)105
- c)110
- d)120
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Cubes of same size are arranged in the form of cube on a table then a ...
Required no. of cubes = 125 – (4 × 5)
= 125 – 20 = 105
= 125 – 20 = 105
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Cubes of same size are arranged in the form of cube on a table then a ...
To solve this problem, let's break it down step by step.
Step 1: Understanding the initial arrangement of cubes
- We start with a large cube made up of smaller cubes.
- Each face of the large cube is made up of smaller cubes.
- Since the large cube is formed by arranging smaller cubes, it is a cube made up of cubes.
Step 2: Removing the corner cubes
- We remove a column of five cubes from each of the four corners of the large cube.
- This means that we remove a total of 4 columns, each containing 5 cubes.
Step 3: Identifying the remaining cubes
- After removing the corner cubes, we are left with a smaller cube inside the large cube.
- The smaller cube is formed by removing the corner cubes from each face of the large cube.
- The smaller cube will have one less layer of cubes on each face compared to the large cube.
Step 4: Counting the cubes in the smaller cube
- To find the number of cubes in the smaller cube, we need to determine the number of cubes on each face and then calculate the total.
- Since each face of the smaller cube has one less layer of cubes compared to the large cube, we subtract 1 from each dimension of the large cube to get the dimensions of the smaller cube.
- Let's assume the large cube has dimensions of n x n x n cubes.
- The smaller cube will have dimensions of (n-1) x (n-1) x (n-1) cubes.
Step 5: Calculating the total number of cubes in the smaller cube
- To find the total number of cubes in the smaller cube, we multiply the dimensions of the smaller cube.
- The total number of cubes in the smaller cube is (n-1) x (n-1) x (n-1).
Step 6: Calculating the total number of cubes after removing the column
- Since the smaller cube is formed by removing the corner cubes from each face of the large cube, the total number of cubes in the solid after removing the column is the sum of the cubes in the smaller cube and the cubes in the removed column.
- The total number of cubes after removing the column is (n-1) x (n-1) x (n-1) + 4 x 5.
Step 7: Simplifying the expression
- We can simplify the expression by expanding it.
- (n-1) x (n-1) x (n-1) = n^3 - 3n^2 + 3n - 1
- 4 x 5 = 20
Step 8: Final answer
- The total number of cubes after removing the column is n^3 - 3n^2 + 3n - 1 + 20.
In this problem, the number of cubes in the large cube is not given. Therefore, we cannot determine the exact number of cubes in the solid after removing the column. We can only provide a general formula for calculating the number of cubes based on the dimensions of the large cube.
Step 1: Understanding the initial arrangement of cubes
- We start with a large cube made up of smaller cubes.
- Each face of the large cube is made up of smaller cubes.
- Since the large cube is formed by arranging smaller cubes, it is a cube made up of cubes.
Step 2: Removing the corner cubes
- We remove a column of five cubes from each of the four corners of the large cube.
- This means that we remove a total of 4 columns, each containing 5 cubes.
Step 3: Identifying the remaining cubes
- After removing the corner cubes, we are left with a smaller cube inside the large cube.
- The smaller cube is formed by removing the corner cubes from each face of the large cube.
- The smaller cube will have one less layer of cubes on each face compared to the large cube.
Step 4: Counting the cubes in the smaller cube
- To find the number of cubes in the smaller cube, we need to determine the number of cubes on each face and then calculate the total.
- Since each face of the smaller cube has one less layer of cubes compared to the large cube, we subtract 1 from each dimension of the large cube to get the dimensions of the smaller cube.
- Let's assume the large cube has dimensions of n x n x n cubes.
- The smaller cube will have dimensions of (n-1) x (n-1) x (n-1) cubes.
Step 5: Calculating the total number of cubes in the smaller cube
- To find the total number of cubes in the smaller cube, we multiply the dimensions of the smaller cube.
- The total number of cubes in the smaller cube is (n-1) x (n-1) x (n-1).
Step 6: Calculating the total number of cubes after removing the column
- Since the smaller cube is formed by removing the corner cubes from each face of the large cube, the total number of cubes in the solid after removing the column is the sum of the cubes in the smaller cube and the cubes in the removed column.
- The total number of cubes after removing the column is (n-1) x (n-1) x (n-1) + 4 x 5.
Step 7: Simplifying the expression
- We can simplify the expression by expanding it.
- (n-1) x (n-1) x (n-1) = n^3 - 3n^2 + 3n - 1
- 4 x 5 = 20
Step 8: Final answer
- The total number of cubes after removing the column is n^3 - 3n^2 + 3n - 1 + 20.
In this problem, the number of cubes in the large cube is not given. Therefore, we cannot determine the exact number of cubes in the solid after removing the column. We can only provide a general formula for calculating the number of cubes based on the dimensions of the large cube.
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Cubes of same size are arranged in the form of cube on a table then a column of five cubes is removed from each of the four cubes is removed from each of the four corners. All the exposed faces of rest of the cube are coloured yellow. Then how many small cubes are there in the solid after the removal of the column?a)100b)105c)110d)120Correct answer is option 'B'. Can you explain this answer?
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