A community library installs a cube-shaped storage unit for archive boxes. The volume of the unit is \(V = 512\) cm\(^{3}\). Work out the side length of the...
Question 1 Report
A community library installs a cube-shaped storage unit for archive boxes. The volume of the unit is \(V = 512\) cm\(^{3}\).
- Work out the side length of the unit, showing that it is a whole number of centimetres. (2)
- A smaller unit has side length equal to \(\sqrt[3]{64}\) cm. Work out this side length. (1)
- Work out the ratio of the volumes of the larger unit to the smaller unit, giving your answer in the form \(n:1\). (2)
- The library needs a total volume of at least 1000 cm\(^{3}\) using only smaller units. Find the least number of smaller units required. (2)
Answer Details
The side of a cube is the cube root of its volume, found here by recognising \(512\) as a perfect cube; the ratio of two cube volumes with a common unit simplifies like any ratio, and a "how many are needed" question always rounds up, since a fractional extra unit still requires a whole one.
- \(8^3=512\), so the side length is \(8\) cm, a whole number. [2 marks]
- \(\sqrt[3]{64}=4\) cm, since \(4^3=64\). [1 mark]
- The volumes are \(8^3=512\) cm\(^3\) and \(4^3=64\) cm\(^3\); ratio \(=512:64=8:1\) (dividing both by \(64\)). [2 marks]
- \(1000\div64=15.625\) smaller units [1 mark]; since a fraction of a unit cannot be used, the least number of whole units required is \(16\) (rounding up). [1 mark]
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