Density Mass Volume

 

We are calculating the volume, so by covering up the V we can see from the triangle above that we have to divide m by d.

 

Before we can do this, however, we have to make sure that we have the correct units. The mass is in kilograms, but the density is in grams per cubic centimetre. This means that we have to first convert the kilograms into grams before proceeding.

2 kg = 2000 g

 

Therefore, the volume of the olive oil can be calculated as follows:

 

\text{Volume} = 2000 \div 0.925 = 2162cm^3

 

 

We are calculating the  mass, so by covering up the m we can see from the triangle above that we have to multiply d by V.

 

However, we don’t know the volume, but we do know that the shape is a cube with a side length of 7 m, so the volume of the cube is:

7 \times 7 \times 7 = 343 m^3

 

Now that we know the volume, we can multiply it by the density in order to calculate the mass:

 

\text{Mass } = 343 \times 10,800,000 = 3,704,400,000 kg

To calculate the answer here, we need to recall the formula:

 

\text{ density} = \text{ mass} \div \text{ volume}

 

In this question, the mass is 2460kg and the volume is 1.2 m^3, so we simply need to substitute these values into the formula as follows:

 

\text{ Density} = 2460 kg \div \, 1.2 m^3 = 2050 kg/m^3

a) In order to calculate the overall volume of the block, we need to add the volume of metal A and the volume of metal B.  Although we don’t have the volume of either metal, we have been given their masses and densities, so we can calculate the volume of each metal accordingly.

 

By rearranging the density formula, or using the triangle, we can work out how to calculate the volume:

\text{density} = \text{ mass}\div \text{ volume}

So:

\text{volume} = \text{ mass}\div \text{ density}

 

The volume of metal A can be calculated as follows:

1200 g \div \, 5 g/cm^3 = 240 cm^3

 

The volume of metal B can be calculated as follows:

600 g \div \, 3 g/cm^3 = 200 cm^3

 

Therefore, if metal A has a volume of 240 cm^3 and B has a volume of 200 cm^3, then their combined volume is simply:

240 cm^3 + 200 cm^3 = 440 cm^3

 

b) As we know from question a), the newly-formed block has a volume of 440 cm^3.

We know that the mass of metal A was 1200 g and the mass of metal B was 600 g, so mass of the block is:

1200 g + \, 600 g = 1800 g

 

The density of this block can be calculated by dividing the mass by the volume as follows:

1800 g \div \, 440 cm^3 = 4.09 g/cm^3

This is quite a challenging question with a lot of calculations going on.

 

Since we have been given the mass of metal C and the ratio of metal A and metal B in metal C, we can therefore calculate the mass of metal A and metal B.  If the ratio of metal A to metal B is 3 : 7, that means that \frac{3}{10} of the mass of metal C comes from metal A and the remaining \frac{7}{10} is metal B.  (We are dealing in tenths here since the sum of the ratio is 10.)

 

The mass of metal A can be calculated as follows:

2500 g \times \dfrac{3}{10} = 750 g

 

The mass of metal B can be calculated as follows:

2500 g   \times \dfrac{7}{10} = 1750 g

 

We now know both the mass and density of both metals A and B, meaning we can work out their respective volumes.

Since

\text{density} = \text{ mass} \div \text {volume}

then

\text{volume} = \text{ mass} \div \text {density}

 

The volume of metal A can be calculated as follows:

750 g \div \, 3.2 g/cm^3 = 234.375 cm^3

 

The volume of metal A can be calculated as follows:

1750 g \div \, 5.5 g/cm^3 = 318.18 cm^3

 

If metal A has a volume of 234.375 cm^3 and metal B has a volume of 318.18 cm^3, then their combined volume is the volume of metal C.

Volume of metal C = 234.375 + 318.18 = 552.5568 cm^3

 

We now know both the mass and volume of metal C, so we are now able to calculate its density.

 

Density of metal C = 2500 g \div \, 552.5568 cm^3 = 4.5 g/cm^3