How many solid iron rods, each of length 7 m and diameter 2 cm, can be manufactured from 0.88 cubic metre of iron?

Introduction / Context:
This problem combines the concept of volume of a cylinder with unit conversion between cubic metres and cubic centimetres. Each iron rod is cylindrical in shape, and many such rods are made from a fixed total volume of molten iron. The task is to determine how many identical rods can be formed without any wastage of material.

Given Data / Assumptions:

• Total volume of iron available = 0.88 m^3.
• Each rod is a solid cylinder.
• Length of each rod L = 7 m.
• Diameter of each rod = 2 cm, so radius r = 1 cm.
• The rods are assumed to be perfectly formed with no loss of volume.

Concept / Approach:
The volume of a cylinder is given by:
Volume of one rod = π * r^2 * h Here r is in centimetres and h must also be in centimetres if we want the volume in cubic centimetres. Since the total volume is given in cubic metres, we convert that to cubic centimetres using:
1 m = 100 cm, so 1 m^3 = 100^3 = 1000000 cm^3. After obtaining both volumes in the same unit, we divide the total volume by the volume of one rod to get the number of rods.

Step-by-Step Solution:
Step 1: Convert the total volume to cubic centimetres. Total volume = 0.88 m^3 = 0.88 * 1000000 = 880000 cm^3. Step 2: Convert the length of one rod to centimetres. Length L = 7 m = 700 cm. Step 3: Radius r = 1 cm, so volume of one rod = π * 1^2 * 700 = 700 * π cm^3. Step 4: Number of rods = total volume / volume of one rod. Number of rods = 880000 / (700 * π). Step 5: Using π = 22 / 7, volume of one rod = 700 * 22 / 7 = 2200 cm^3. Step 6: Number of rods = 880000 / 2200 = 400.

Verification / Alternative check:
We can simplify 880000 / 2200 by cancelling zeros and factors. Dividing numerator and denominator by 100 gives 8800 / 22, which equals 400. This confirms the calculation and shows that the result is an exact integer, so the given data are consistent and there is no fractional rod.

Why Other Options Are Wrong:
300 rods would use only 300 * 2200 = 660000 cm^3 of iron, leaving a large unused volume.
350 rods would correspond to 770000 cm^3, still less than the given total volume.
450 rods would require 990000 cm^3, which is more iron than available.
500 rods would need 1100000 cm^3, again exceeding the given 880000 cm^3 of iron.

Common Pitfalls:
Learners often forget to convert all dimensions to the same unit system, mixing metres with centimetres. Another frequent error is using the diameter instead of the radius in the cylinder formula or misapplying the factor of π. Rounding π too early can also lead to slight discrepancies, but here with π = 22 / 7 the answer comes out exactly as an integer. Carefully tracking units and using the correct formulas avoids these mistakes.

Final Answer:
The number of solid iron rods that can be made is 400 rods.