Two traffic cones used to close a bus lane are mathematically similar. The smaller cone has height 6 cm and volume 45 cm\(^3\). The larger cone has height 1...
Question 1 Report
Two traffic cones used to close a bus lane are mathematically similar. The smaller cone has height 6 cm and volume 45 cm\(^3\). The larger cone has height 18 cm. The council orders both sizes and wants the larger cone's volume for a materials quote. Both sizes are stored together in the same equipment shed.
Work out the volume of the larger cone. (4)
Answer Details
The two traffic cones are mathematically similar, so their linear dimensions scale by the same factor, but volume (a 3-dimensional quantity) scales by the cube of that linear scale factor.
\[ \text{Linear scale factor} = \frac{18}{6} = 3 \]\[ \text{Volume scale factor} = 3^3 = 27 \]\[ \text{Volume of larger cone} = 45 \times 27 = 1215 \text{ cm}^3 \]The volume of the larger cone is \(1215\) cm\(^3\). [4] (1 for the linear scale factor 3, 1 for cubing it to get 27, 1 for multiplying \(45 \times 27\), 1 for the final value 1215)
The most common error with similar solids is applying the linear scale factor directly to the volume (giving \(45 \times 3 = 135\)); always cube the linear scale factor first when scaling a volume, and square it when scaling an area.
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