two metal rods A and B have their initial length in the ratio 2:3 and coefficients of linear expansion in the ratio of 4:3. when they are heated through same temperature difference the ratio of their linear expansion is

Explanation

Here's how to solve this problem and find the ratio of their linear expansions:1. Define variables:Let L_A and L_B be the initial lengths of rods A and B, respectively.Let α_A and α_B be the coefficients of linear expansion of rods A and B, respectively.Let ΔT be the same temperature difference for both rods.Let ΔL_A and ΔL_B be the linear expansions of rods A and B, respectively.2. Use the formula for linear expansion:ΔL = α * L * ΔT3. Apply the information given:L_A / L_B = 2 / 3 (ratio of initial lengths)α_A / α_B = 4 / 3 (ratio of coefficients of linear expansion)ΔT is the same for both rods (given in the problem)4. Find the ratio of linear expansions (ΔL_A / ΔL_B):Substitute the given information into the formula for linear expansion for both rods:ΔL_A = α_A * L_A * ΔTΔL_B = α_B * L_B * ΔTDivide both equations:(ΔL_A / ΔL_B) = (α_A * L_A * ΔT) / (α_B * L_B * ΔT)Since ΔT cancels out and the ratio of coefficients remains 4/3, we only need to consider the ratio of initial lengths:(ΔL_A / ΔL_B) = (L_A / L_B) * (α_A / α_B)(ΔL_A / ΔL_B) = (2/3) * (4/3)5. Simplify the equation:(ΔL_A / ΔL_B) = 8/9