A box room measures \(4\) m long, \(3\) m wide and \(2\) m high, as shown in the diagram. A removals firm wants to know the longest straight rod that will f...
Question 1 Report
A box room measures \(4\) m long, \(3\) m wide and \(2\) m high, as shown in the diagram. A removals firm wants to know the longest straight rod that will fit inside the room.
Work out the length of the diagonal shown. Give your answer correct to 3 significant figures. (3)
Answer Details
The diagram shows the long diagonal of a cuboid, running from one bottom corner to the opposite top corner. This diagonal is found in one step using the three-dimensional version of Pythagoras' theorem, which extends the two-dimensional case to three mutually perpendicular edges meeting at a corner.
\[ d = \sqrt{4^2 + 3^2 + 2^2} \] [1 mark]
\[ d = \sqrt{16 + 9 + 4} = \sqrt{29} \] [1 mark]
\[ d = 5.39 \text{ m (3 s.f.)} \] [1 mark]
An equivalent method finds the diagonal of the floor first, \(\sqrt{4^2+3^2}=5\) m, and then treats that floor diagonal and the \(2\) m height as the two legs of a second right-angled triangle: \(\sqrt{5^2+2^2}=\sqrt{29}\), the same result.
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