Question

A copper cube of side length \(40 . \mathrm{cm}\) is heated from \(20 .{ }^{\circ} \mathrm{C}\) to \(120{ }^{\circ} \mathrm{C} .\) What is the change in the volume of the cube? The linear expansion coefficient of copper is \(17 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\)

Short answer

Answer: The change in volume of the copper cube is approximately 106.196 cm³.

Step-by-step solution

Calculate the change in temperature

To find the change in temperature, simply subtract the initial temperature from the final temperature. In this case, the initial temperature is \(20^\circ \mathrm{C}\) and the final temperature is \(120^\circ \mathrm{C}\): \(\Delta T = T_{final} - T_{initial} = 120 - 20 = 100^\circ \mathrm{C}\)

Calculate the change in side length

To calculate the change in side length, use the linear expansion formula, which is given by: \(\Delta L = L_{initial}\cdot\alpha\cdot\Delta T\) Where: - \(\Delta L\) is the change in length - \(L_{initial}\) is the initial length - \(\alpha\) is the linear expansion coefficient - \(\Delta T\) is the change in temperature Plug in the given values for the side length, linear expansion coefficient, and the change in temperature: \(\Delta L = 40\,\mathrm{cm} \cdot (17\cdot 10^{-6}\,^{\circ} \mathrm{C}^{-1}) \cdot (100^{\circ} \mathrm{C})\) \(\Delta L \approx 0.068\,\mathrm{cm}\)

Calculate the final side length

Add the change in side length to the initial side length to find the final side length: \(L_{final} = L_{initial} + \Delta L = 40\,\mathrm{cm} + 0.068\,\mathrm{cm} \approx 40.068\,\mathrm{cm}\)

Calculate the initial and final volumes

Use the side lengths to calculate the initial and final volumes of the cube, using the formula: \(V = L^3\) Initial volume: \(V_{initial} = (40\,\mathrm{cm})^3 = 64000\,\mathrm{cm}^3\) Final volume: \(V_{final} = (40.068\,\mathrm{cm})^3 \approx 64106.196\,\mathrm{cm}^3\)

Calculate the change in volume

Subtract the initial volume from the final volume to find the change in volume: \(\Delta V = V_{final} - V_{initial} = 64106.196\,\mathrm{cm}^3 - 64000\,\mathrm{cm}^3 \approx 106.196\,\mathrm{cm}^3\) The change in volume of the copper cube is approximately \(106.196\,\mathrm{cm}^3\).