QUESTION 15 3.8 km / sec ? __ mile / year (Use scientific notation; 50000 = 5e4) [1 km = 0.6214 mile]
The Correct Answer and Explanation is:
Answer: 7.4e7 mile/year
Explanation
To solve this problem, a conversion of units from kilometers per second to miles per year is required. This process involves changing both the unit of distance (from kilometers to miles) and the unit of time (from seconds to years). The most reliable method for this is dimensional analysis, which uses conversion factors to systematically cancel out old units and introduce new ones.
Step 1: Identify the Given Information and Goal
- Initial Rate: 3.8 km/sec
- Distance Conversion Factor: 1 km = 0.6214 mile
- Goal: Convert the rate to miles/year
Step 2: Set Up the Time Conversion Factors
To convert seconds to years, a chain of conversions is needed. The standard time equivalences are:
- 60 seconds = 1 minute
- 60 minutes = 1 hour
- 24 hours = 1 day
- 365.25 days = 1 year (Using 365.25 accounts for leap years, providing a more accurate average.)
Step 3: Construct the Full Conversion Equation
The initial rate is multiplied by a series of conversion factors. Each factor is a ratio equal to one (e.g., 0.6214 mi / 1 km = 1). The factors are arranged so that the unwanted units cancel out, leaving only miles in the numerator and years in the denominator.
The complete setup is:
(3.8 km / 1 sec) × (0.6214 mile / 1 km) × (60 sec / 1 min) × (60 min / 1 hr) × (24 hr / 1 day) × (365.25 day / 1 yr)
Step 4: Calculation and Scientific Notation
Multiplying the numerical values gives the final answer:
3.8 × 0.6214 × 60 × 60 × 24 × 365.25 = 74,489,425.392
The last step is to express this number in scientific notation and apply the correct number of significant figures. The initial measurement, 3.8 km/sec, has two significant figures. Therefore, the final answer must also be rounded to two significant figures.
74,489,425.392 ≈ 74,000,000
In scientific notation, this is written as 7.4 x 10⁷. Using the requested “e” format, the final answer is 7.4e7.thumb_upthumb_down
