Aluminum has a density of 2.99 grams per cubic cm. and a specific heat of 0.22 calories per gram degree Celsius. A cube of aluminum is sitting out in the sun where it absorbs 16.2 calories of heat and warms from 62*F to 149*F How many inches wide are the sides of the cube?
To find the width of the sides of the aluminum cube, we need to determine the side length of the cube first. We can use the information provided to calculate the change in temperature and the amount of heat absorbed by the aluminum cube.
1. Convert the initial and final temperatures from Fahrenheit to Celsius:
- Initial temperature: 62 °F = (62 - 32) / 1.8 = 16.7 °C
- Final temperature: 149 °F = (149 - 32) / 1.8 = 65.0 °C
2. Calculate the change in temperature:
- ΔT = Final temperature - Initial temperature
= 65.0 °C - 16.7 °C
= 48.3 °C
3. Calculate the amount of heat absorbed by the aluminum cube (Q):
- Q = Specific heat * Mass * ΔT
= (0.22 calories/g °C) * Mass * 48.3 °C
= 16.2 calories
4. Rearrange the equation to solve for Mass:
- Mass = Q / (Specific heat * ΔT)
= 16.2 calories / (0.22 calories/g °C * 48.3 °C)
= 16.2 / (0.22 * 48.3) g
= 0.5108 g
5. Now, let's consider the density of aluminum:
- Density = Mass / Volume
- Since the cube has equal side lengths, the volume is given by:
Volume = (Side length)³
Rearrange the equation to solve for Side length:
- Side length = (Mass / Density)^(1/3)
- Side length = (0.5108 g / 2.99 g/cm³)^(1/3)
- Side length = 0.8577 cm
6. Convert the side length from cm to inches:
- 1 cm = 0.3937 inches
- Side length (in inches) = 0.8577 cm * 0.3937 inches/cm
- Side length ≈ 0.3379 inches
Therefore, the sides of the aluminum cube are approximately 0.3379 inches wide.