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A metal rod A of length 25 cm expands by 0.050 cm when heated from 0 deg C to 100 deg C. A rod B of a different metal of length 40 cm expands by 0.040 cm for the same temperature rise. A third rod C of total length 50 cm is made of pieces of A and B joined end to end, and it expands by 0.030 cm when heated from 0 deg C to 50 deg C. Find the length of each portion in the composite rod.

1. A = 10 cm, B = 40 cm
2. A = 40 cm, B = 10 cm
3. A = 25 cm, B = 25 cm
4. A = 20 cm, B = 30 cm

Correct answer: A = 10 cm, B = 40 cm

Solution

From rod A: alpha_A = 0.050/(25*100) = 2*10⁻⁵ per deg C. From rod B: alpha_B = 0.040/(40*100) = 1*10⁻⁵ per deg C. For composite rod C of 50 cm over a 50 deg C rise, the total expansion is (L_A*alpha_A + L_B*alpha_B)*50 = 0.030 cm with L_A + L_B = 50. Solving gives L_A = 10 cm and L_B = 40 cm.

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