Video: AQA GCSE Mathematics Higher Tier Pack 3 • Paper 3 • Question 20

The density of lead is 11.34 g/cm³. Given that 1 km = 2.2 lbs and 1 cm = 0.4 inches, calculate the density of lead in pounds per cubic inch.

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Video Transcript

The density of lead is 11.34 grams per centimetre cubed. Given that one kilogram is equal to 2.2 pounds and one centimetre equals 0.4 inches, calculate the density of lead in pounds per cubic inch.

Before starting this problem, it is worth remembering the formula density is equal to mass divided by volume. The density of lead in the question was given in grams per centimetre cubed. This is because mass can be measured in grams and volume in centimetres cubed.

We want to convert this into pounds per cubic inch. This can be written using the same notation as grams per centimetre cubed. The mass can be measured in pounds and the volume in inches cubed. In this case, we’re converting from metric units to imperial units. We need to convert 11.34 grams per centimetre cubed into pounds per inch cubed.

We were told in the question the conversion that one kilogram is equal to 2.2 pounds. As there are 1000 grams in one kilogram, then 1000 grams must also be equal to 2.2 pounds. We now need to scale this conversion to calculate how many pounds are in 11.34 grams.

Dividing both sides by 1000 shows us that one gram is equal to 0.0022 pounds. Multiplying the left-hand side by 11.34 gives us 11.34 grams. And multiplying the left-hand side by 11.34 gives us 0.024948. Therefore, 11.34 grams is equal to 0.024948 pounds.

We now need to consider the volume. Once again, we were told in the question that one centimetre is equal to 0.4 inches. As this is a measure of length, we need to cube both sides to get a measure of volume, as volume is measured in centimetres cubed or inches cubed. One cubed is equal to one, and 0.4 cubed is equal to 0.064. Therefore, one centimetre cubed is equal to 0.064 inches cubed.

We can therefore say that 11.34 grams per centimetre cubed is equal to 0.024948 pounds per 0.064 inches cubed. We want to calculate the density in pounds per cubic inch or per inch cubed. Therefore, we need to divide both sides by 0.064. 0.024948 divided by 0.064 is equal to 0.3898125. This means that the density of lead is 0.3898125 pounds per inch cubed or per cubic inch.

It would be sensible to round this to two decimal places. The nine is the deciding number. And as this is bigger than five, we round up. The density of lead to two decimal places is 0.39 pounds per cubic inch.

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