Iron has a density of 7.86 g/cm³ (1 cm³ = 1 mL)

Iron has a density of 7.86 g/cm³ (1 cm³ = 1 mL). Calculate the volume (in dL) of a piece of iron having a mass of 4.53 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:

Answer: 5.76 dL

Step-by-Step Explanation:

We are given:

• Density of iron = 7.86 g/cm³
• Mass of iron = 4.53 kg
• 1 cm³ = 1 mL
• We want volume in dL (deciliters)

Step 1: Convert the density to kg/dL

We start by converting 7.86 g/cm³ to kg/dL.

Since:

• 1 g = 0.001 kg
• 1 cm³ = 1 mL = 0.01 dL (since 1 dL = 100 mL)

So:Density=7.86 g/cm3=7.86 gmL=7.86×0.001 kg0.01 dL=0.00786 kg0.01 dL=0.786 kgdL\text{Density} = 7.86 \, \text{g/cm}^3 = 7.86 \, \frac{\text{g}}{\text{mL}} = \frac{7.86 \times 0.001 \, \text{kg}}{0.01 \, \text{dL}} = \frac{0.00786 \, \text{kg}}{0.01 \, \text{dL}} = 0.786 \, \frac{\text{kg}}{\text{dL}}Density=7.86g/cm3=7.86mLg​=0.01dL7.86×0.001kg​=0.01dL0.00786kg​=0.786dLkg​

Step 2: Use the density formula to find volume

The formula for density is:Density=MassVolume\text{Density} = \frac{\text{Mass}}{\text{Volume}}Density=VolumeMass​

Solving for volume:Volume=MassDensity=4.53 kg0.786 kg/dL=5.76 dL\text{Volume} = \frac{\text{Mass}}{\text{Density}} = \frac{4.53 \, \text{kg}}{0.786 \, \text{kg/dL}} = 5.76 \, \text{dL}Volume=DensityMass​=0.786kg/dL4.53kg​=5.76dL

Final Answer:

5.76 dL\boxed{5.76 \, \text{dL}}5.76dL​

Explanation

To find the volume of a piece of iron with a given mass, we must use the relationship between mass, volume, and density. The basic formula is:Density=MassVolume\text{Density} = \frac{\text{Mass}}{\text{Volume}}Density=VolumeMass​

Rearranging this formula to find the volume:Volume=MassDensity\text{Volume} = \frac{\text{Mass}}{\text{Density}}Volume=DensityMass​

However, before applying the formula, the units of the given values must be consistent. In this case, the mass is provided in kilograms (kg), the density is given in grams per cubic centimeter (g/cm³), and the volume must be found in deciliters (dL). Because 1 cm³ is equivalent to 1 milliliter (mL) and 100 mL make up 1 dL, we know that 1 cm³ = 0.01 dL. Also, 1 gram equals 0.001 kilograms.

To convert the density from 7.86 g/cm³ to kg/dL, we multiply the numerator (grams) by 0.001 to convert to kilograms and divide the denominator (cm³) by 100 to convert to dL:7.86 g1 cm3=7.86×0.001 kg1×0.01 dL=0.786 kg/dL\frac{7.86 \, \text{g}}{1 \, \text{cm}^3} = \frac{7.86 \times 0.001 \, \text{kg}}{1 \times 0.01 \, \text{dL}} = 0.786 \, \text{kg/dL}1cm37.86g​=1×0.01dL7.86×0.001kg​=0.786kg/dL

Now that the units match, we can plug the mass of 4.53 kg into the volume formula:Volume=4.53 kg0.786 kg/dL=5.76 dL\text{Volume} = \frac{4.53 \, \text{kg}}{0.786 \, \text{kg/dL}} = 5.76 \, \text{dL}Volume=0.786kg/dL4.53kg​=5.76dL

This result represents the volume of the piece of iron.