The diagram shows a rectangular field measuring 240 m by 175 m. (a) One hectare is 10 000 m\(^2\). Calculate the area of the field, in hectares. [2] (b) A m...

Both parts are unit conversions, but they differ in an important way: part (a) converts an area and part (b) converts a volume through a scale factor. The rule is that a length scale factor \(k\) becomes \(k^2\) for areas and \(k^3\) for volumes.

(a) The area of the field in square metres is

\(240 \times 175 = 42\,000\) m\(^2\) [M1]

Since one hectare is \(10\,000\) m\(^2\), divide by \(10\,000\):

\(42\,000 \div 10\,000 = 4.2\) hectares [A1]

(b) The scale \(1 : 50\) is a length ratio, so every length on the building is 50 times the corresponding length on the model. Volume involves three lengths, so the volume scale factor is \(50^3 = 125\,000\):

\(640 \times 50^3 = 80\,000\,000\) cm\(^3\) [M1]

Converting to cubic metres uses the same cubing idea: \(1\) m \(= 100\) cm, so \(1\) m\(^3\) \(= 100^3 = 1\,000\,000\) cm\(^3\). Hence

\(80\,000\,000 \div 1\,000\,000 = 80\) m\(^3\) [A1]

Multiplying the model volume by 50 rather than \(50^3\) gives \(32\,000\) cm\(^3\), which is the single most common error on scale questions. Whenever a ratio is applied, ask whether the quantity is a length, an area or a volume, and raise the ratio to the matching power.