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Question

The speed of light is approximately $3 	imes 10^8$ m/s.  If a star is 4 light-years away, what is its distance in kilometers? (1 light-year = distance light travels in one year)

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: Option C
## CONCEPT FRAMEWORK
	**Primary Concept**: Distance Calculation Using Speed and Time
	**Related Topics**: Units Conversion, Scientific Notation
	**Prerequisites**: Understanding of speed, time, distance relationship, basic unit conversions (meters to kilometers, seconds to years).
## SOLUTION PATHWAY
**Given Information**:
	Speed of light: $3 	imes 10^8$ m/s
	Distance: 4 light-years
	1 light-year = distance light travels in one year
**Method & Steps**:
	1. Calculate the distance light travels in one year in meters.
		Reasoning: We need to find the distance for 1 light year, so we multiply the speed of light by the number of seconds in a year.
		Formula/Rule:  $distance = speed 	imes time$
	2. Calculate the number of seconds in a year.
		Working: 1 year = 365.25 days, 1 day = 24 hours, 1 hour = 3600 seconds.
		Key Point: We use 365.25 days to account for leap years.
	3. Calculate the distance light travels in one year in meters
		Working: $1 	ext{ year} = 365.25 	imes 24 	imes 3600 	ext{ seconds} = 31557600 	ext{ seconds}$
		 Distance in 1 year $= (3 	imes 10^8 	ext{ m/s}) 	imes 31557600 	ext{ s} = 9.46728 	imes 10^{15} 	ext{ m}$
	4. Calculate the distance in meters for 4 light-years.
		Working: Distance in 4 light-years $= 4 	imes 9.46728 	imes 10^{15} 	ext{ m} = 3.786912 	imes 10^{16} 	ext{ m}$
	5. Convert the distance from meters to kilometers.
		Working: $1 	ext{ km} = 1000 	ext{ m}$.  So, we divide the distance in meters by 1000.
		Key Point: Conversion from smaller unit (meter) to larger unit (km) involves division.
	6. Final Distance in Kilometers
		Result: Distance in 4 light-years $= (3.786912 	imes 10^{16} 	ext{ m}) / 1000 = 3.786912 	imes 10^{13} 	ext{ km} \approx 3.8 	imes 10^{13} 	ext{ km}$
		Verification: The final answer is within the range of the options given, and the units are correct (kilometers).
 ## OPTION ANALYSIS
[A] :x: This option is too small.  It is likely a result of errors in the calculation or unit conversion.
[B] :x: This option is also too small. It likely resulted from incorrect power calculation or not converting from meters to km.
[C] ✓: This option is the correct distance, derived from accurate calculations and unit conversions.
[D] :x: This option is significantly too small. It likely resulted from not multiplying by the correct time or incorrect units.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Incorrect unit conversions (meters to kilometers). Always double check unit conversions.
		2.[Error 2]: Errors in calculating the number of seconds in a year.  Remember to include the number of days, hours and seconds in a year.
	**Quick Tips**:
		1.[Helpful shortcut]: Remember that 1 light-year is approximately $10^{16}$ meters. If asked to find the distance of multiple light years, you just need to multiply the number of light years by this value in meters, then convert to km.
		2.[Memory aid]: Always write out your steps clearly, especially when dealing with unit conversions and large numbers in scientific notation.

NAGA SAI · Answer

$3.8 	imes 10^{10}$ km

NAGA SAI · Answer

$1.2 	imes 10^{13}$ km

NAGA SAI · Answer

The correct Option is "$3.8 	imes 10^{13}$ km"

NAGA SAI · Answer

$1.2 	imes 10^9$ km

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