According to legend, the following challenge led Archimedes to the discovery of

Question

According to legend, the following challenge led Archimedes to the discovery of his famous principle: Hieron, king of Syracuse, was suspicious that a new crown that he had received from the royal goldsmith was not pure gold, as claimed. Archimedes was ordered to determine whether the crown was in fact made of pure gold, with the condition that only a nondestructive test would be allowed. Rather than answer the problem in the politically most expedient way (or perhaps extract a bribe from the goldsmith), Archimedes thought about the problem scientifically. The legend relates that when Archimedes stepped into his bath and caused it to overflow, he realized that he could answer the challenge by comparing the volume of water displaced by the crown with the volume of water displaced by an amount of pure gold equal in weight to the crown. If the crown was made of pure gold, the two volumes would be equal. If some other (less dense) metal had been substituted for some of the gold, then the crown would displace more water than the pure gold.

Explanation / Answer

Part A: if density is D, and "apparent weight" (call it Wapp) is the weight of the crown in water, then if we say the crown has a volume V: (let Dw be the density of water). By Archimedes' principle, the crown will be buoyed up by a force equal to the weight of water which it displaces. Wapp = D*V*g - DwV*g Dividing by the actual weight (Wact = D*V*g) you get: Wapp/Wact = 1-Dw/D Part A.1: Solve the equation above for Wact: Wact = Wapp/(1-Dw/D) Part A.2 Not exactly why this is asked, since we kind of used it to get the other answers, but here's something: Wapp = (1-Dw/D)*Wact Hope this helps.

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