This is a test post that shows how we can present problems step by step. Each step starts as a collapsed accordion; the student first reads the problem statement, and if they can’t solve it on their own, they open the steps one by one. After each step is opened, they see a “Next Step” button that automatically opens and scrolls to the next step.
A Real Problem from Chapter 1
A metal coin has a mass of 9.8 grams. When it is placed in a graduated cylinder filled with water, the water level rises from 20 cc to 21.4 cc. What is the density of the coin’s metal? Based on this value, can you guess what metal the coin is made of?
Hint
Remember two things:
– Density means mass per unit volume: $rho = dfrac{m}{V}$
– Volume of the coin = increase in water volume (water displacement method — Archimedes’ principle)
Finding the Volume of the Coin
The volume of the coin is equal to the increase in water volume in the cylinder:
$$V = 21.4 – 20 = 1.4 text{ cm}^3$$
(Reminder: 1 cc = 1 cubic centimeter)
Choosing the Formula and Substitution
From the definition of density:
$$rho = frac{m}{V} = frac{۹/۸ text{ g}}{۱/۴ text{ cm}^3}$$
Calculation
$$rho = ۷ text{ g/cm}^3$$
Therefore, the density of the coin’s metal is 7 grams per cubic centimeter.
Final Answer and Interpretation
The density of the coin is $rho = 7 text{ g/cm}^3$.
By comparing this with the densities of known metals, this value is very close to the density of iron (7.87) and zinc (7.14). Therefore, the coin is most likely made of one of these two metals or an alloy containing them.
Conceptual Note: Density is a “fingerprint” for a material. If you measure the density of an unknown sample accurately, you can make good guesses about its composition — the same method geologists use to identify rocks.
The exercise above was a simple example of step-by-step problem solving. In future lessons, you will encounter more complex problems using this same format. See you there! 👋
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