Math  /  Numbers & Operations

QuestionPractice--Space Calculations.
1. Convert the following into km. a. 52.1 AU

Studdy Solution

STEP 1

What is this asking? We need to change a distance from astronomical units (AU) to kilometers (km). Watch out! Don't mix up AU and km!
AU is a *massive* unit of distance used in astronomy, while km is used for distances on Earth.

STEP 2

1. Define the conversion factor.
2. Perform the conversion.

STEP 3

One astronomical unit (AU) is the average distance between the Earth and the Sun, which is approximately **149.6 million km**.
So, we can write this relationship as:
1 AU=149.6106 km1 \text{ AU} = 149.6 \cdot 10^6 \text{ km}

STEP 4

This equation is our **key** to converting between AU and km.
It tells us how many kilometers are packed into a *single* AU!

STEP 5

We're given a distance of **52.1 AU**, and we want to express this distance in km.
We can use our conversion factor from the previous step.

STEP 6

We want to multiply by one, because that doesn't change the value of our distance, just the units.
We can write one as:
149.6106 km1 AU\frac{149.6 \cdot 10^6 \text{ km}}{1 \text{ AU}}

STEP 7

Now, let's **multiply** our given distance by this clever form of one:
52.1 AU149.6106 km1 AU52.1 \text{ AU} \cdot \frac{149.6 \cdot 10^6 \text{ km}}{1 \text{ AU}}

STEP 8

Notice how the units of AU in the numerator and denominator add to zero, leaving us with only km.
This is *exactly* what we wanted!

STEP 9

Now, let's **crunch the numbers**:
52.1149.6106 km=7794.16106 km52.1 \cdot 149.6 \cdot 10^6 \text{ km} = 7794.16 \cdot 10^6 \text{ km}

STEP 10

We can express this in scientific notation as **7.794161097.79416 \cdot 10^9 km**.
That's a *huge* distance!

STEP 11

The distance of 52.1 AU is equal to 7.794161097.79416 \cdot 10^9 km.

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