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Answer :
Final answer:
The density of a material given in kg/m³ is converted to a new set of units by first converting to g/cm³ and then adjusting for the new units of mass and volume. The resulting density, when converted to units of 50 g and 15,625 cm³, is approximately 16.384 g in the new units, leading us to option C as the correct answer. Option (C) is correct.
Explanation:
The question involves converting the density of a material from one set of units to another. The given density is 128 kg/m³ in SI units, and we need to find its numerical value when the unit of length is 25 cm and the unit of mass is 50 g.
To convert the density, we should first convert the given density to g/cm³, which is a more commonly used unit for solids and liquids and allows us to work with the smaller units we're interested in. Since 1 kg is equal to 1000 g and 1 m³ is equal to 1,000,000 cm³, the density in g/cm³ is calculated by taking 128 kg/m³ and multiplying it by (1000 g/kg) / (1,000,000 cm³/m³), which gives us a density of 0.128 g/cm³.
Now, to convert this to the new units, where the unit of length is 25 cm (or 0.25 m), we cube this length to get the new unit of volume: 0.25 m x 0.25 m x 0.25 m = 0.015625 m³, which is equivalent to 15,625 cm³. Hence, the volume ratio is 15,625 cm³/new unit volume = 1,000,000 cm³/m³. Now, for the mass of 50 g, which is the mass unit here, we must divide the g/cm³ value by this ratio. The resulting density is 0.128 g/cm³ (1000 g/kg) / (15,625 cm³/new unit volume) = 0.128 x 1000 / 15,625.
After calculating, we get a new numerical value of density of 8.192 g in the new mass unit per new volume unit. Since the mass unit is actually 50 g, we further reduce this by a factor of 50, resulting in a numerical value of 8.192/50 = 0.16384. Finally, to get the answer in the whole number, we multiply by 100 (since the original value was in kg and we converted to g). The final answer is 0.16384 x 100 = 16.384, which rounded to the nearest whole number gives us 16, corresponding to option C.
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