Question

Express this product in units of \(\mathrm{km}^{3}\) with the appropriate number of significant figures: \((3.2 \mathrm{km}) \times(4.0 \mathrm{m}) \times\) \(\left(13 \times 10^{-3} \mathrm{mm}\right)\)

Short answer

Answer: \(1.7\times10^{-7}\,\mathrm{km}^3\)

Step-by-step solution

Convert all units to kilometers

First, we need to convert all units to kilometers. We have the conversion factors: 1 \(\mathrm{km}\) = 1000 \(\mathrm{m}\), and 1 \(\mathrm{m}\) = 1000 \(\mathrm{mm}\) The given values are: 3.2 \(\mathrm{km}\) 4.0 \(\mathrm{m}\) \(\left(13 \times 10^{-3}\right) \mathrm{mm}\) Convert the meters and millimeters to kilometers: \((4.0\,\mathrm{m}) \times \frac{1\,\mathrm{km}}{1000\,\mathrm{m}} = 0.004\,\mathrm{km}\) \((13 \times 10^{-3}\,\mathrm{mm})\times\frac{1\,\mathrm{m}}{1000\,\mathrm{mm}}\times\frac{1\,\mathrm{km}}{1000\,\mathrm{m}} = 1.3\times10^{-8}\,\mathrm{km}\)

Multiply the values

Now, multiply the converted values: \((3.2\,\mathrm{km})\times(0.004\,\mathrm{kim})\times(1.3\times10^{-8}\,\mathrm{km})\)

Apply significant figures

We need to consider the appropriate number of significant figures for the product. In multiplication, we use the same number of significant figures as the value with the least significant figures in the calculation. The given values have 2, 2, and 2 significant figures respectively. So, the product should have 2 significant figures. \((3.2\,\mathrm{km})\times(0.004\,\mathrm{km})\times(1.3\times10^{-8}\,\mathrm{km})\) = \((3.2\times0.004\times1.3)\times10^{-8}\,\mathrm{km}^3\) Calculate the product and round it to 2 significant figures: \((3.2\times0.004\times1.3) = 0.01664\), rounded to 0.017 (2 significant figures) \(0.017\times10^{-8}\,\mathrm{km}^3 = 1.7\times10^{-7}\,\mathrm{km}^3\) The final answer is \(1.7\times10^{-7}\,\mathrm{km}^3\).