Iron has a density of 7.86 g/cm3. Calculate the volume (in dL ) of a piece of iron having a mass of 4.84 kg. Note that the density is provided in different units of volume and mass than the desired units of volume (dL) and the given units of mass (kg). You will need to express the density in kg/dL(1 cm3=1 mL) before calculating the volume for the piece of iron. Express your answer to three significant figures.
The Correct Answer and Explanation is :
To calculate the volume of a piece of iron, we can use the formula:
[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
]
Step 1: Conversion of Density to the Required Units
Given that the density of iron is 7.86 g/cm³, we need to express this in kg/dL because the mass is provided in kg, and the desired volume units are in dL.
We know the following unit conversions:
- 1 cm³ = 1 mL (since 1 cm³ = 1 mL)
- 1 mL = 0.001 L
- 1 L = 10 dL
- 1 kg = 1000 g
Therefore, we first need to convert the density from g/cm³ to kg/dL:
- Start by converting grams to kilograms. Since there are 1000 grams in a kilogram, the density in kg/cm³ is:
[
7.86 \, \text{g/cm³} = \frac{7.86}{1000} \, \text{kg/cm³} = 0.00786 \, \text{kg/cm³}
]
- Now, convert cm³ to dL. Since 1 cm³ = 0.001 L and 1 L = 10 dL, 1 cm³ = 0.01 dL. Therefore, the density in kg/dL is:
[
0.00786 \, \text{kg/cm³} \times 0.01 \, \text{cm³/dL} = 0.0000786 \, \text{kg/dL}
]
Thus, the density of iron is ( 0.0000786 \, \text{kg/dL} ).
Step 2: Calculate the Volume
The mass of the piece of iron is 4.84 kg, and the density we just calculated is 0.0000786 kg/dL.
Now we can use the formula:
[
\text{Volume} = \frac{\text{Mass}}{\text{Density}} = \frac{4.84 \, \text{kg}}{0.0000786 \, \text{kg/dL}} = 61609.63 \, \text{dL}
]
Step 3: Express the Answer to Three Significant Figures
To express this answer to three significant figures, the volume is approximately:
[
\boxed{61600 \, \text{dL}}
]
Conclusion
The volume of the piece of iron with a mass of 4.84 kg and a density of 7.86 g/cm³ is approximately 61600 dL when rounded to three significant figures. The conversion process involved converting the density to the appropriate units of kg/dL before applying the formula for volume.