Question

The distance from Earth to the sun is 1 astronomical unit $(\mathrm{AU})$ ) or $1.496\times {10}^9$ kilometers. If Mercury is $0.39\mathrm{AU}$ from the sun, how many kilometers from the sun is it? A. $10.79\times {10}^6$ kilometers B. $5.8\times {10}^9$ kilometers C. $1.079\times {10}^7$ kilometers D. $5.8\times {10}^8$ kilometers

Short answer

D. 5.8 x 10^8 kilometers

Step-by-step solution

Identify the given values

The problem provides the distance from Earth to the sun as 1 astronomical unit (AU) or 1.496 x 10^9 kilometers. It also provides the distance from Mercury to the sun as 0.39 AU.

Set up the equation

To determine the distance from Mercury to the sun in kilometers, we use the relation: $$\text{Distance (Mercury)}=\text{Distance (Earth-Sun in km)}\times \text{Distance (Mercury-Sun in AU)}$$ Substituting the provided values: $$\text{Distance (Mercury)}=1.496\times {10}^9\text{ km}\times 0.39\text{ AU}$$

Perform the multiplication

Calculate the result by multiplying: $$1.496\times {10}^9\times 0.39$$ Simplify the multiplication: $$1.496\times 0.39\times {10}^9=0.58344\times {10}^9$$

Convert to proper scientific notation

The result from the previous step is 0.58344 x 10^9 km. To convert it into proper scientific notation: $$0.58344\times {10}^9=5.8344\times {10}^8$$ Round this to the appropriate number of significant figures based on the given options, which are to two significant figures: $$\text{Distance (Mercury)}=5.8\times {10}^8\text{ km}$$

Match with the options

The calculated distance 5.8 x 10^8 km matches with the provided option D.

Key concepts

Distance Measurement

In astronomy, distance measurement is crucial to understanding the vast universe. One key unit is the Astronomical Unit $(AU)$, representing the average distance from Earth to the Sun. It equals approximately 1.496 x 10^9 kilometers (km). This unit simplifies comparing distances within our solar system. For example, Mercury is 0.39 AU away from the Sun, meaning Mercury's distance from the Sun is 0.39 times that of the Earth's distance. To convert this distance into kilometers, multiply the given AU value by the AU's kilometer equivalent. Understanding how to measure distances in AU and convert them is fundamental in solving related problems.

Scientific Notation

Scientific notation is a way to express very large or very small numbers conveniently. It uses powers of 10 to make these numbers more manageable. For instance, 1.496 x 10^9 in scientific notation represents 1,496,000,000. The format is: $$a\times {10}^b$$ where $a$ is a number between 1 and 10, and $b$ is an integer indicating how many places to move the decimal point.

In our exercise, both the distance from Earth to the Sun (1.496 x 10^9 km) and the calculated distance for Mercury (5.8 x 10^8 km) are in scientific notation. This simplifies calculations and ensures clarity when working with such large values. Properly converting results into scientific notation after performing operations is essential for accuracy and precision.

Multiplication

Multiplication is a basic yet crucial arithmetic operation used in various calculations. When dealing with units of measurement and scientific notation, it helps determine precise values. In our exercise, we need to multiply the Earth's distance to the Sun by Mercury's distance in AU: $$\text{Distance (Mercury)}=1.496\times {10}^9\text{ km}\times 0.39\text{ AU}$$ Performing this multiplication involves combining coefficients (1.496 and 0.39) and powers of 10 (10^9). Thus:

1. Multiply the coefficients: 1.496 x 0.39 = 0.58344
2. Keep the power of 10 intact: 0.58344 x 10^9
3. Convert to proper scientific notation: 0.58344 x 10^9 = 5.8344 x 10^8

Multiplying values correctly, especially in scientific notation, ensures accurate measurement conversion and solution.