The difference between the outer and inner radii of a hollow right circular cylinder of length 14 cm is 1 cm. If the volume of the metal used in making the cylinder is 176 cm$^3$, find the outer and inner radii of the cylinder.
Generated by claude-sonnet-4-6 · 2026-06-15 10:36 · grounding rag
Model Answer
Let outer radius = $R$ cm, inner radius = $r$ cm.
Given:
- $R - r = 1$ … (i)
- Volume of metal = $\pi(R^2 - r^2) \times h = 176$ cm³
Substituting $h = 14$ cm and $\pi = \dfrac{22}{7}$:
$$\frac{22}{7} \times (R^2 - r^2) \times 14 = 176$$
$$44(R^2 - r^2) = 176$$
$$R^2 - r^2 = 4$$
$$(R+r)(R-r) = 4$$
Since $R - r = 1$:
$$R + r = 4 \quad \text{…(ii)}$$
Adding (i) and (ii): $2R = 5 \Rightarrow R = 2.5$ cm
Subtracting (i) from (ii): $2r = 3 \Rightarrow r = 1.5$ cm
Outer radius = 2.5 cm, Inner radius = 1.5 cm
Source: Chapter 12, Section 12.3 — Volume of a Combination of Solids
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Explanation
- The key formula is: Volume of hollow cylinder = $\pi(R^2 - r^2)h$, which uses the difference of squares.
- Examiners expect you to set up two equations from the two given conditions and solve simultaneously — show both equations clearly.
- Use $\pi = \dfrac{22}{7}$ as instructed and simplify step by step to avoid errors.