Question

question 7 (6 points) according to the factor unit method, when converting from 7.0km to meters (7.0km = ? m), the correct way to set up the equation and most correct solution is:

○ a
7.0km x \\(\frac{1000m}{1km}\\) = 7.000 x \\(10^3\\)m

○ b
7.0km x \\(\frac{1000m}{1km}\\) = 7.0 x \\(10^3\\)m

○ c
7.0km x \\(\frac{1km}{0.1m}\\) = 7m

○ d
7.0km x \\(\frac{1m}{1000km}\\) = 7.0 x \\(10^{-4}\\)m

○ e
7.0km x \\(\frac{1000m}{1km}\\) = 7.00 x \\(10^3\\)m

○ f
7.0km x \\(\frac{1000m}{1km}\\) = 7 x \\(10^3\\)m

○ g
7.0km x \\(\frac{1000m}{1km}\\) = 7km

○ h
7.0km x \\(\frac{1km}{0.1m}\\) = 7.00m

Explanation:

Step1: Recall unit conversion factor

We know that \(1\,\text{km} = 1000\,\text{m}\), so the conversion factor is \(\frac{1000\,\text{m}}{1\,\text{km}}\) (to convert km to m, we multiply by this factor to cancel out km units).

Step2: Analyze each option

- Option a: \(7.0\,\text{km} \times \frac{1000\,\text{m}}{1\,\text{km}} = 7000\times 10^{3}\,\text{m}\) is wrong. \(7.0\times1000 = 7.0\times 10^{3}\), not \(7000\times 10^{3}\).
- Option b: \(7.0\,\text{km} \times \frac{1000\,\text{m}}{1\,\text{km}} = 7.0\times 10^{3}\,\text{m}\) is correct in value but let's check significant figures and other options. Wait, let's check option e: \(7.0\,\text{km} \times \frac{1000\,\text{m}}{1\,\text{km}} = 7.00\times 10^{3}\,\text{m}\)? Wait, no, \(7.0\times1000 = 7000 = 7.0\times 10^{3}\) or \(7.00\times 10^{3}\)? Wait, \(7.0\) has two significant figures. When we multiply \(7.0\times1000\), the result should have two significant figures? Wait, no, \(1000\) here is an exact conversion factor (defined, so infinite significant figures). So \(7.0\times1000 = 7000\), which can be written as \(7.0\times 10^{3}\) (two significant figures) or \(7.00\times 10^{3}\) (three)? Wait, no, \(7.0\) has two, so the result should have two. Wait, but let's check the options again. Wait, option e: \(7.0\,\text{km} \times \frac{1000\,\text{m}}{1\,\text{km}} = 7.00\times 10^{3}\,\text{m}\) – no, \(7.0\times1000 = 7000 = 7.0\times 10^{3}\) (two sig figs) or \(7.00\times 10^{3}\) would be three, but \(7.0\) has two. Wait, maybe the question is about the correct setup and the correct number of significant figures. Wait, the original value is \(7.0\,\text{km}\) (two significant figures). So when we convert, \(7.0\,\text{km} \times \frac{1000\,\text{m}}{1\,\text{km}} = 7.0\times 10^{3}\,\text{m}\) (option b) or \(7.00\times 10^{3}\)? Wait, no, \(1000\) is exact, so the number of significant figures is determined by \(7.0\), which is two. Wait, but option e is \(7.00\times 10^{3}\), which is three sig figs, which is incorrect. Wait, maybe I made a mistake. Wait, \(7.0\) km: the zero is significant, so two sig figs. So \(7.0\times1000 = 7000\), which can be written as \(7.0\times 10^{3}\) (two sig figs) or \(7000.\) (but that's not common). Wait, let's check the options again. Option b: \(7.0\times 10^{3}\), option e: \(7.00\times 10^{3}\), option f: \(7\times 10^{3}\). Wait, the correct setup is multiplying by \(\frac{1000\,\text{m}}{1\,\text{km}}\). Now, let's check the calculation: \(7.0\,\text{km} \times \frac{1000\,\text{m}}{1\,\text{km}} = 7.0\times 1000\,\text{m} = 7.0\times 10^{3}\,\text{m}\) (option b) or \(7000\,\text{m}=7.0\times 10^{3}\,\text{m}\). Wait, but option e is \(7.00\times 10^{3}\), which would be if the original was \(7.00\,\text{km}\), but it's \(7.0\). Wait, maybe the question is considering that \(1000\) has four significant figures? No, \(1000\) as a conversion factor is exact (1 km is exactly 1000 m), so it doesn't affect significant figures. So the number of significant figures is determined by \(7.0\), which is two. So option b: \(7.0\times 10^{3}\) is correct? Wait, no, wait \(7.0\times1000 = 7000\), which is \(7.0\times 10^{3}\) (two sig figs) or \(7.00\times 10^{3}\) would be three, but that's incorrect. Wait, maybe the answer is option b? Wait, no, let's check the options again. Wait, option e: \(7.0\,\text{km} \times \frac{1000\,\text{m}}{1\,\text{km}} = 7.00\times 10^{3}\,\text{m}\) – no, \(7.0\times1000 = 7000 = 7.0\times 10^{3}\) (two decimal places? No, significant figures). Wait, maybe the question is not about significant figures but about the correct setup and calculation. The correct conversion factor is \(\frac{1000\,\text{m}}{1\,\text{km}}\). So multiplying \(7.0\,\text{km}\) by that gives \(7.0\times 1000\,\text{m}=7.0\times 10^{3}\,\text{m}\) (option b) or \(7000\,\text{m}=7.0\times 10^{3}\,\text{m}\). Wait, but option e is \(7.00\times 10^{3}\), which is three sig figs, but \(7.0\) has two. Wait, maybe the original problem has \(7.0\) (two sig figs), so the answer should have two. So option b: \(7.0\times 10^{3}\,\text{m}\) is correct. Wait, but let's check the options again. Option b: \(7.0\,\text{km} \times \frac{1000\,\text{m}}{1\,\text{km}} = 7.0\times 10^{3}\,\text{m}\) – yes, that's correct. Option e: \(7.00\times 10^{3}\) – no, because \(7.0\) has two sig figs. Option f: \(7\times 10^{3}\) – one sig fig, incorrect. So the correct option is b? Wait, no, wait \(7.0\times1000 = 7000 = 7.0\times 10^{3}\) (two sig figs) or \(7.00\times 10^{3}\) would be three, but that's not correct. Wait, maybe the question is considering that \(1000\) is written as \(1.000\times 10^{3}\), so \(7.0\times1.000\times 10^{3}=7.0\times 10^{3}\) (option b) or \(7.00\times 10^{3}\) (option e). Wait, no, \(7.0\) has two sig figs, so the result should have two. So option b is correct. Wait, but let's check the calculation again. \(7.0\,\text{km} \times \frac{1000\,\text{m}}{1\,\text{km}} = 7.0\times 1000\,\text{m} = 7000\,\text{m} = 7.0\times 10^{3}\,\text{m}\) (since \(7000 = 7.0\times 10^{3}\) when considering two sig figs). So option b is correct.

Answer:

b. \(7.0\,\text{km} \times \frac{1000\,\text{m}}{1\,\text{km}} = 7.0\times 10^{3}\,\text{m}\)