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Difficulty: HardExponents, Roots, and Scientific Notation

An astronomical unit (AU) is a unit of length equal to approximately 1.50×1081.50 \times 10^8 kilometers. A research probe travels through space at a constant speed of 2.50×1042.50 \times 10^4 meters per second. Which of the following is closest to the number of hours it will take the probe to travel a distance of 5.005.00 AU?

  1. A
    8.33×1008.33 \times 10^0
  2. 8.33×1038.33 \times 10^3Answer
  3. C
    8.33×1048.33 \times 10^4
  4. D
    3.00×1073.00 \times 10^7
  5. E
    1.08×10111.08 \times 10^{11}

Answer

The correct answer is 8.33×1038.33 \times 10^3 hours.
To find the travel time in hours, the distance of 5.005.00 AU is first converted to meters: 5.00×1.50×108 km×103 m/km=7.50×1011 m5.00 \times 1.50 \times 10^8\text{ km} \times 10^3\text{ m/km} = 7.50 \times 10^{11}\text{ m}. Dividing this distance by the speed of 2.50×104 m/s2.50 \times 10^4\text{ m/s} yields a time of 3.00×107 seconds3.00 \times 10^7\text{ seconds}. Dividing this time by the 3,600 seconds3,600\text{ seconds} in an hour results in 3.00×1073,600≈8,333.33 hours\frac{3.00 \times 10^7}{3,600} \approx 8,333.33\text{ hours}, which is written in standard scientific notation as 8.33×103 hours8.33 \times 10^3\text{ hours}.

Step-by-Step Solution

1
Convert the total distance from astronomical units (AU) to meters.
Distance = 7.50×10117.50 \times 10^{11} meters
Since 1 AU≈1.50×108 km1\text{ AU} \approx 1.50 \times 10^8\text{ km} and 1 km=103 m1\text{ km} = 10^3\text{ m}, then 1 AU≈1.50×1011 m1\text{ AU} \approx 1.50 \times 10^{11}\text{ m}. Therefore, a distance of 5.00 AU=5.00×(1.50×1011 m)=7.50×1011 m5.00\text{ AU} = 5.00 \times (1.50 \times 10^{11}\text{ m}) = 7.50 \times 10^{11}\text{ m}.
2
Calculate the travel time in seconds.
Time = 3.00×1073.00 \times 10^7 seconds
Dividing the total distance in meters by the speed in meters per second gives: 7.50×1011 m2.50×104 m/s=3.00×107 seconds\frac{7.50 \times 10^{11}\text{ m}}{2.50 \times 10^4\text{ m/s}} = 3.00 \times 10^7\text{ seconds}.
3
Convert the travel time from seconds to hours.
Time ≈8.33×103\approx 8.33 \times 10^3 hours
Since 1 hour=3,600 seconds1\text{ hour} = 3,600\text{ seconds} (which is 3.60×103 seconds3.60 \times 10^3\text{ seconds}), we divide the time in seconds by 3,6003,600: 3.00×1073.60×103≈0.833×104=8.33×103 hours\frac{3.00 \times 10^7}{3.60 \times 10^3} \approx 0.833 \times 10^4 = 8.33 \times 10^3\text{ hours}.

Key Concept

Applying division and unit conversion with numbers written in scientific notation.
Estimated Time:2m 0s
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Topics

Exponents, Roots, and Scientific Notation

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