A volume of 66 cubic centimetres of silver is drawn into a uniform wire 1 mm in diameter. What will be the length of the wire in metres?

Introduction / Context:
This problem is similar to other wire drawing questions, where a known volume of metal is reshaped into a cylindrical wire. The key idea is that the volume of metal remains constant during the reshaping, so the volume of the wire equals the given volume. Knowing the diameter, we can use the cylinder volume formula to find the length of the wire.

Given Data / Assumptions:

• Volume of silver V = 66 cm^3.
• Wire is cylindrical and uniform.
• Diameter of wire = 1 mm.
• We want the wire length in metres.
• No loss of material occurs during drawing.

Concept / Approach:
The volume of a cylinder is:
V = π * r^2 * h, where r is radius in centimetres and h is length in centimetres when we want volume in cubic centimetres. The diameter is given in millimetres, so we must convert to centimetres before using the formula. After finding h in centimetres, we convert it to metres.

Step-by-Step Solution:
Step 1: Convert diameter to radius in centimetres. Diameter = 1 mm = 0.1 cm. Radius r = 0.1 / 2 = 0.05 cm. Step 2: Use V = π * r^2 * h with V = 66 cm^3. 66 = π * (0.05)^2 * h. Step 3: Compute r^2 = (0.05)^2 = 0.0025. So 66 = π * 0.0025 * h. Step 4: Solve for h. h = 66 / (π * 0.0025). Rewrite 0.0025 as 1 / 4000, so π * 0.0025 = π / 4000. Then h = 66 / (π / 4000) = 66 * 4000 / π = 264000 / π cm. Step 5: Use π ≈ 22 / 7. h ≈ 264000 * 7 / 22 = (264000 / 22) * 7. 264000 / 22 = 12000, so h ≈ 12000 * 7 = 84000 cm. Step 6: Convert centimetres to metres. h = 84000 cm = 84000 / 100 = 840 m? This intermediate suggests rechecking. A simpler exact approach gives h = 26400 / π cm, and with π ≈ 22 / 7, h = 26400 * 7 / 22 = 8400 cm. Thus correct h = 8400 cm = 84 m.

Verification / Alternative check:
Using the form h = 26400 / π cm and taking π ≈ 3.14, we get h ≈ 26400 / 3.14 ≈ 8417 cm, which is close to 8400 cm. The small difference arises from rounding π. Therefore, the length of about 84 m is consistent with the given volume and diameter.

Why Other Options Are Wrong:
90 m is slightly larger than the correct value and would give a volume larger than 66 cm^3 when substituted back into the formula.
168 m and 336 m are much too large and would correspond to volumes several times the given 66 cm^3.
42 m is half of 84 m and would yield a volume only half of what is required.

Common Pitfalls:
As with similar problems, unit conversion between millimetres, centimetres and metres is a common source of error. Squaring the radius and handling the small decimal 0.05 requires care. Some learners also forget to convert the final length into metres or misplace decimal points when doing so. Working patiently with fractions and checking results using an approximate value of π helps ensure accuracy.

Final Answer:
The length of the wire formed is 84 metres.