A community library sells two geometrically similar globe-shaped bookends, made from the same solid material. The smaller globe has diameter \(8\) cm and ma...
Question 1 Report
A community library sells two geometrically similar globe-shaped bookends, made from the same solid material. The smaller globe has diameter \(8\) cm and mass \(0.9\) kg. The larger globe has diameter \(12\) cm, as shown.
- Find the linear scale factor from the smaller globe to the larger globe. (2)
- Work out the mass of the larger globe, correct to 3 significant figures. (3)
- The larger globe costs \(\pounds4\) per kg to produce, and the library has a \(\pounds12\) budget per globe. State, with a reason, whether the budget is enough. (3)
Answer Details
Because the two globes are geometrically similar and made from the same material, their masses scale with the cube of the ratio of their diameters, the same rule used for volumes of similar solids since mass is proportional to volume when density is constant.
(a) The linear scale factor is the ratio of diameters: \(12\div8=1.5\). [2 marks]
(b) The mass scale factor is the cube of the linear scale factor: \(1.5^{3}=3.375\). Multiplying the smaller globe's mass: \(0.9\times3.375=3.0375\), which rounds to \(3.04\) kg (3 s.f.). [3 marks]
(c) Using the more accurate, unrounded mass, the production cost is \(3.0375\times4=\pounds12.15\), which is more than the library's \(\pounds12\) budget per globe, so the budget is NOT enough. [3 marks]
Using the unrounded mass \(3.0375\) kg in part (c), rather than the rounded \(3.04\) kg, matters here because the cost, \(\pounds12.15\), sits so close to the \(\pounds12\) budget limit that early rounding could easily flip the conclusion.
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