The diameter of a brass ring at 30 °C is 50.0 cm. To what temperature must this ring be heated to increase its…

Analyze the question: A brass ring expands when heated. We know its original diameter, its new diameter, and its linear expansivity, and we need the final temperature.

Concept: Linear expansion is given by

ΔL = L₀ · α · Δθ

so Δθ = ΔL / (L₀ · α)

Solution:

• L₀ = 50.0 cm
• L = 50.29 cm
• ΔL = 50.29 − 50.0 = 0.29 cm
• α = 1.9 × 10⁻⁵ K⁻¹

Δθ = 0.29 / (50.0 × 1.9 × 10⁻⁵)

Denominator: 50.0 × 1.9 × 10⁻⁵ = 9.5 × 10⁻⁴

Δθ = 0.29 / (9.5 × 10⁻⁴) = 305.26 K

Final temperature = 30 + 305.26 = 335.26 °C ≈ 335.3 °C

Answer: The ring must be heated to about 335.3 °C, which is option D.

Option C (306.1 °C) is a common trap. It comes from finding only the temperature change (about 305 K) and forgetting to add the starting temperature of 30 °C.

Tip: In thermal expansion questions, always ask yourself, "Is the question asking for the change in temperature or the final temperature?" If it asks "to what temperature," add the change to the initial temperature.