Final answer:
The change in length of the copper pipeline when the temperature varies between 46°F and 109°F is approximately 21.665 mm. This is calculated using the coefficient of linear expansion for copper and the formula for thermal expansion.
Step-by-step explanation:
The question is regarding the thermal expansion of a copper pipeline that carries hot water. As the temperature of the water (and the pipeline) varies, the length of the copper pipeline will also change due to thermal expansion. We will use the formula for linear expansion ΔL = αL₀ΔT, where ΔL is the change in length, α is the coefficient of linear expansion for copper, L₀ is the original length of the material, and ΔT is the change in temperature.
To find the corresponding changes in the length of the pipeline, we will need to convert the temperatures from degrees Fahrenheit to degrees Celsius, calculate the change in temperature (ΔT), and then apply the formula using the coefficient of linear expansion for copper, which is approximately 16.5 × 10⁻¶ (1/°C).
First, convert the temperatures from °F to °C using the formula: T(°C) = (T(°F) - 32) × 5/9. For 46°F, we get T(°C) = (46 - 32) × 5/9 ≈ 7.778°C, and for 109°F, we get T(°C) = (109 - 32) × 5/9 ≈ 42.778°C. The change in temperature (ΔT) is then 42.778°C - 7.778°C = 35°C.
Using the formula ΔL = αL₀ΔT with the values α = 16.5 × 10⁻¶ (1/°C), L₀ = 38 m, and ΔT = 35°C, we calculate the change in length as: ΔL = 16.5 × 10⁻¶ × 38 m × 35°C ≈ 0.021665 m or 21.665 mm.
Therefore, the corresponding change in the length of the copper pipeline when the temperature varies between 46°F and 109°F is approximately 21.665 mm.