The diameter of our solar system (depending on how you define “solar system”) is about 8,980,000,000,000 kilometers. The distance from our solar system to the nearest star, Proxima Centauri, is about 39,900,000,000,000 kilometers. A light-year is the distance light travels in one year, and it is equal to about 9,460,000,000,000 kilometers.

a) Write each of these numbers in scientific notation. Explain how you got your answers, and explain why you know that your method works.

b) You’d like to know both the end-to-end distance within the solar system and the distance to Proxima Centauri in light-years. Using your scientific notation numbers, convert these distances. What is the ratio of the solar system’s length to the Proxima Centauri distance?

c) Light travels at 300,000 kilometers per second. Write this rate in scientific notation. When Proxima Centauri sends out light at that speed, how long does it take to reach our solar system? Can you convert to minutes, hours, days, or years? Does your answer in years correspond to the distance in light-years from our solar system to Proxima Centauri? Why or why not?

To convert a number to scientific notation, we express it as a decimal number multiplied by a power of 10. The power of 10 indicates the number of times the decimal point is moved.

- Diameter of the solar system:

8,980,000,000,000 kilometers can be written as 8.98 x 10¹² km.

- Distance to Proxima Centauri:

39,900,000,000,000 kilometers can be written as 3.99 x 10¹³ km.

- One light-year:

9,460,000,000,000 kilometers can be written as 9.46 x 10¹² km.

b) Converting distances to light-years:

Solar system's length: 8.98 x 10¹² km

Dividing this value by the distance in one light-year, 9.46 x 10¹² km, we get:

(8.98 x 10¹² km) / (9.46 x 10¹² km) = 0.951

The solar system's length is approximately 0.951 light-years.

Distance to Proxima Centauri: 3.99 x 10¹³km

Dividing this value by the distance in one light-year, 9.46 x 10¹² km, we get:

(3.99 x 10¹³ km) / (9.46 x 10¹² km) ≈ 4.22

The distance to Proxima Centauri is approximately 4.22 light-years.

The ratio of the solar system's length to the distance to Proxima Centauri is:

0.951 light-years / 4.22 light-years ≈ 0.225

c) Light speed in scientific notation:

300,000 kilometers per second can be written as 3 x 10⁵ km/s.

To convert this to minutes, hours, days, or years, we need to know the conversion factors:

- 60 seconds in a minute

- 60 minutes in an hour

- 24 hours in a day

- 365 days in a year

To calculate the time it takes for light from Proxima Centauri to reach our solar system:

Distance: 4.22 light-years

Time = Distance / Speed = 4.22 light-years / (3 x 10⁵ km/s)

Converting light-years to kilometers using the conversion factor of 9.46 x 10¹² km per light-year:

= (4.22 x 9.46 x 10¹² km) / (3 x 10⁵ km/s)

= 12.648 x 10¹² km / 3 x 10⁵ km/s

= 42.16 x 10⁶ s

= 4.216 x 10⁷ s

To convert to minutes, hours, days, or years, we divide the total seconds by the appropriate conversion factors:

- 60 seconds in a minute

- 60 minutes in an hour

- 24 hours in a day

- 365 days in a year (approximate value)

Let's convert the time in seconds to years:

= 4.216 x 10⁷ s/(60 s/min * 60 min/h * 24 h/day * 365 day/yr)

= 1.34 years

The answer in years does correspond to the distance in light-years from our solar system to Proxima Centauri because light travels at a constant speed. One light-year is defined as the distance light travels in one year. Therefore, the time it takes for light to travel from Proxima Centauri to our solar system is approximately 1.34 years, which matches the distance of 4.22 light-years.

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