The size of an iron block is 15 cm × 10 cm × 20 cm. How many such blocks will…
2023
The size of an iron block is 15 cm × 10 cm × 20 cm. How many such blocks will be needed for the construction of a solid iron cube of minimum size?
- A.
90
- B.
72
- C.
120
- D.
60
Show answer & explanation
Correct answer: B
Step-by-Step Calculation
1. Determine the side length of the minimum cube:
For the iron blocks (15 cm x 10 cm x 20 cm) to perfectly form a cube, the side length of that cube must be a common multiple of the block's dimensions. To find the minimum size, we calculate the LCM of 15, 10, and 20.
Prime factorization:
15 = 3 * 5
10 = 2 * 5
20 = 2^2 * 5
LCM = 2^2 * 3 * 5 = 60 cm.
Therefore, the smallest cube that can be formed has a side length of 60 cm.
2. Calculate the number of blocks required:
Volume of the cube = 60 cm * 60 cm * 60 cm = 216,000 cubic cm.
Volume of one iron block = 15 cm * 10 cm * 20 cm = 3,000 cubic cm.
Number of blocks = Volume of cube / Volume of one block
Number of blocks = 216,000 / 3,000 = 72.