A metallic cylindrical pipe of uniform thickness has volume 748 cubic centimetres. Its length is 14 cm and its external radius is 9 cm. What is the thickness of the pipe?

Introduction / Context:
This question involves the volume of a hollow cylinder, which represents a cylindrical pipe made of metal. The pipe has an external radius, an internal radius, and a given length. The metal occupies the region between the outer and inner cylindrical surfaces. By using the formula for the volume of a hollow cylinder, we can determine the inner radius and therefore the thickness of the pipe. Such problems are useful for understanding how material quantities relate to dimensions in engineering and manufacturing contexts.

Given Data / Assumptions:
• Volume of the metal in the pipe is 748 cubic centimetres.
• Length (height) of the pipe, h, is 14 cm.
• External radius, R, is 9 cm.
• Internal radius, r, is unknown and must be found.
• Thickness of the pipe is R - r in centimetres.
• We assume π is taken as 22/7 or left as π; the result is exact either way because the numbers are chosen to simplify.

Concept / Approach:
The volume of a hollow cylinder is given by V = π (R² - r²) h, where R is the outer radius, r is the inner radius, and h is the height or length. We know V, R, and h, so we can first solve for (R² - r²). Once we determine r², we can take the square root to obtain r. Finally, the thickness t of the pipe is simply the difference between the external and internal radii, t = R - r. The calculations are straightforward because the values have been selected to give a perfect square for r².

Step-by-Step Solution:
Step 1: Write the hollow cylinder volume formula: V = π (R² - r²) h. Step 2: Substitute the known values: 748 = π (9² - r²) * 14. Step 3: Compute 9² = 81, so we have 748 = 14π (81 - r²). Step 4: Divide both sides by 14π to isolate (81 - r²): (81 - r²) = 748 / (14π). Step 5: If we take π = 22/7, then 14π = 14 * 22/7 = 44, so 748 / 44 = 17. Step 6: Therefore, 81 - r² = 17, which gives r² = 81 - 17 = 64. Step 7: Hence r = √64 = 8 cm. The thickness t is R - r = 9 - 8 = 1 cm.

Verification / Alternative check:
We can verify by recomputing the volume using the found inner radius. With R = 9 cm, r = 8 cm, and h = 14 cm, the volume is V = π (R² - r²) h = π (81 - 64) * 14 = π * 17 * 14. Using π = 22/7, we get V = 22/7 * 17 * 14. First 14 / 7 = 2, so this is 22 * 17 * 2 = 44 * 17 = 748 cubic centimetres, which matches the given volume. This confirms that the computed inner radius and the resulting thickness of 1 cm are correct.

Why Other Options Are Wrong:
0.5 cm and 1.5 cm would give inner radii of 8.5 cm or 7.5 cm respectively, which would produce different volume values when placed in the formula and would not equal 748 cubic centimetres.
2 cm and 2.5 cm correspond to inner radii of 7 cm and 6.5 cm, which again would lead to a much larger metal volume than specified. Substituting these into V = π (R² - r²) h quickly shows mismatches with the given volume.

Common Pitfalls:
A common mistake is to treat 748 cubic centimetres as the volume of a solid cylinder rather than a hollow one, which leads to ignoring the inner cavity and using the formula V = πR²h directly. Another error is to forget that R² - r² must be positive and mis-handle subtraction. Some learners also misapply π approximations, especially when dividing by 14π. Careful substitution, stepwise algebra, and a final verification by recomputing the volume help avoid these errors.

Final Answer:
The thickness of the metallic cylindrical pipe is 1 cm.