A water company compares two cube-shaped reservoir designs. Reservoir A has an edge length of \(2^{3}\) m, and Reservoir B has an edge length of \(2^{2}\) m...
Question 1 Report
A water company compares two cube-shaped reservoir designs. Reservoir A has an edge length of \(2^{3}\) m, and Reservoir B has an edge length of \(2^{2}\) m.
- Work out the edge length of Reservoir A, in metres. (1)
- Write down the volume of Reservoir A as a single power of 2. (2)
- Work out the volume of Reservoir A, in cubic metres, as an ordinary number. (1)
- Write down the volume of Reservoir B as a single power of 2. (2)
- Work out the volume of Reservoir B, in cubic metres. (1)
- The company can only afford to build one reservoir, and will choose the one with the larger volume. State, with a reason, which reservoir should be built. (1)
Answer Details
This question finds the volume of two cubes with edge lengths given as powers of 2, then compares them to decide which reservoir to build.
(a) Evaluating the power: \[ 2^3 = 8 \text{ m} \] [1 mark]
(b) The volume of a cube is its edge length cubed, and raising a power to a power multiplies the exponents: \[ \left(2^3\right)^3 = 2^{3 \times 3} = 2^9 \] [2 marks]
(c) Evaluating the power gives the volume as an ordinary number: \[ 2^9 = 512 \text{ m}^3 \] [1 mark]
(d) Similarly for Reservoir B: \[ \left(2^2\right)^3 = 2^{2 \times 3} = 2^6 \] [2 marks]
(e) Evaluating the power: \[ 2^6 = 64 \text{ m}^3 \] [1 mark]
(f) Comparing the two volumes: Reservoir A holds \(512 \text{ m}^3\) compared with Reservoir B's \(64 \text{ m}^3\), so Reservoir A has the larger volume and should be built. [1 mark]
An edge length one power of 2 larger (\(2^3\) versus \(2^2\)) does not just double the volume; because volume scales with the cube of the edge length, the volume ratio here is \(2^3 = 8\) times, matching \(512 \div 64 = 8\).
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