Question

1. an astronomical unit (au) is used to express great distances in space. it is based upon the distance from earth to the sun. a formula for converting any distance \\(d\\) in miles to au is \\(\text{au} = \frac{d}{93,000,000}\\). alpha centauri is 270,000 au from the sun. how far is that in miles?

\\(2.5 \times 10^{13}\\)

\\(3.44 \times 10^{-3}\\)

\\(3.44 \times 10^{3}\\)

\\(2.5 \times 10^{-13}\\)

Explanation:

Identify the given values and formula

We are given the formula for converting a distance \(d\) in miles to astronomical units (\(\text{AU}\)):

$$\text{AU} = \frac{d}{93,000,000}$$

We are also given that the distance to Alpha Centauri is \(270,000\text{ AU}\). We need to find the distance \(d\) in miles.

Rearrange the formula to solve for the variable

Using the Solving for a Variable knowledge point:

$$d = \text{AU} \times 93,000,000$$

Substitute the given values and calculate

Using the Algebraic Simplification knowledge point:

$$d = 270,000 \times 93,000,000$$

$$d = 25,110,000,000,000$$

Convert the result to scientific notation

To express \(25,110,000,000,000\) in scientific notation, we move the decimal point 13 places to the left:

$$d \approx 2.5 \times 10^{13}\text{ miles}$$

This matches the first option.

Answer:

• (A) \(2.5 \times 10^{13}\) (Correct answer)
• (B) \(3.44 \times 10^{-3}\)
• (C) \(3.44 \times 10^{3}\)
• (D) \(2.5 \times 10^{-13}\)