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Chapter 9: Problem 15

The density of platinum is \(21500 \mathrm{kg} / \mathrm{m}^{3} .\) Find the ratio of the volume of \(1.00 \mathrm{kg}\) of platinum to the volume of $1.00 \mathrm{kg}$ of aluminum.

Short Answer

Expert verified
Answer: The ratio of the volume of 1 kg of platinum to the volume of 1 kg of aluminum is \(\frac{27}{215}\).

Step by step solution

01

Find the volume of 1 kg of Platinum

To find the volume of 1 kg of Platinum, use the density formula, which is: Density = \(\frac{Mass}{Volume}\) Therefore, Volume = \(\frac{Mass}{Density}\) The mass of platinum is 1 kg, and the density is given as \(21500 \mathrm{kg} / \mathrm{m}^{3}.\) So, Volume of platinum = \(\frac{1 \mathrm{kg}}{21500 \mathrm{kg} / \mathrm{m}^{3}} = \frac{1}{21500} \mathrm{m}^{3}\).
02

Find the volume of 1 kg of Aluminum

To find the volume of 1 kg of Aluminum, use the same density formula: Volume = \(\frac{Mass}{Density}\) The mass of aluminum is 1 kg, and the density is given as \(2700 \mathrm{kg} / \mathrm{m}^{3}.\) So, Volume of aluminum = \(\frac{1 \mathrm{kg}}{2700 \mathrm{kg} / \mathrm{m}^{3}} = \frac{1}{2700} \mathrm{m}^{3}\).
03

Calculate the ratio of the volumes

Finally, we'll calculate the ratio of the volume of platinum to the volume of aluminum, by dividing the volume of platinum by the volume of aluminum: Ratio = \(\frac{\text{Volume of platinum}}{\text{Volume of aluminum}} = \frac{\frac{1}{21500} \mathrm{m}^{3}}{\frac{1}{2700} \mathrm{m}^{3}}\) To simplify the expression, multiply the numerator and denominator by the least common multiple (LCM) of 21500 and 2700, which is 58050: Ratio = \(\frac{\frac{1}{21500} \mathrm{m}^{3} \times 58050}{\frac{1}{2700} \mathrm{m}^{3} \times 58050} = \frac{2700}{21500}\).
04

Simplify the ratio

Lastly, simplify the ratio by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 200: Simplified ratio = \(\frac{2700}{21500} = \frac{2700 \div 200}{21500 \div 200} = \frac{27}{215}\). The ratio of the volume of 1 kg of platinum to the volume of 1 kg of aluminum is \(\frac{27}{215}\).

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