Question

The mean distance from the earth to the sun is \(9.3 \times\) \(10^{7} \mathrm{mi} .\) What is this distance in kilometers? in centimeters?

Short answer

The mean distance from the earth to the sun is approximately \(1.49868 \times 10^8 km\) in kilometers and \(1.49868 \times 10^{13} cm\) in centimeters.

Step-by-step solution

Convert miles to kilometers

Firstly, we will convert the distance in miles to kilometers. We are given the mean distance from the earth to the sun in miles: \(9.3 \times 10^7 mi\). To convert this to kilometers, we will use the conversion factor: 1 mile = 1.60934 kilometers. So, the mean distance from the earth to the sun in kilometers would be: \( (9.3 \times 10^7 mi) \times (1.60934 km/mi) \)

Calculate the distance in kilometers

Now, multiply the given distance in miles by the conversion factor to get the distance in kilometers: \( (9.3 \times 10^7 mi) \times (1.60934 km/mi) = 1.49868 \times 10^8 km \) So, the mean distance from the earth to the sun in kilometers is \(1.49868 \times 10^8 km\).

Convert kilometers to centimeters

Next, we will convert the distance in kilometers to centimeters. We already have the mean distance from the earth to the sun in kilometers: \(1.49868 \times 10^8 km\). To convert this to centimeters, we will use the conversion factor: 1 kilometer = 100,000 centimeters. So, the mean distance from the earth to the sun in centimeters would be: \( (1.49868 \times 10^8 km) \times (100,000 cm/km) \)

Calculate the distance in centimeters

Now, multiply the given distance in kilometers by the conversion factor to get the distance in centimeters: \( (1.49868 \times 10^8 km) \times (100,000 cm/km) = 1.49868 \times 10^{13} cm \) So, the mean distance from the earth to the sun in centimeters is \(1.49868 \times 10^{13} cm\). To sum up, the mean distance from the earth to the sun is approximately \(1.49868 \times 10^8 km\) in kilometers and \(1.49868 \times 10^{13} cm\) in centimeters.

Key concepts

Distance Measurement

Distance measurement is a critical concept across various scientific and practical fields. It's the way we determine how far apart two points are in space. Different units are used depending on the scale, such as miles, kilometers, and centimeters. It's essential to understand these units and conversions between them, especially in fields like astronomy and geography. For instance, measuring the distance from the earth to the sun in terms of miles helps scientists understand astronomical scales.

Common distance units include:

• Miles, often used in the United States for road distances and large-scale geographical measurements.
• Kilometers, widely used internationally for similar purposes.
• Centimeters, more suitable for smaller measurements on a personal or lab scale.

Understanding how to convert between these units is vital for accurate scientific calculations.

Scientific Notation

Scientific notation is a method of writing very large or small numbers conveniently. It helps in simplifying calculations and comparisons by using powers of ten. For example, the distance from the earth to the sun can be expressed as \(9.3 \times 10^{7}\) miles.

Scientific notation is beneficial because it:

• Reduces the error margin when dealing with extremely large quantities like astronomical distances.
• Makes it easier to perform multiplication and division with exponential components.
• Simplifies reading and writing of numbers, especially in scientific reports and articles.

This notation is simply a product of a number (between 1 and 10) and 10 raised to an exponent, which turns complex numbers into more manageable forms for calculation and communication.

Miles to Kilometers Conversion

Converting miles to kilometers is an essential skill, especially if you are using international or scientific standards, as the kilometer is the more commonly used unit. To convert a distance in miles to kilometers, you use the conversion factor: 1 mile = 1.60934 kilometers.

The conversion process involves simple multiplication. For example, to convert the mean earth-sun distance from miles to kilometers, use:
\[ (9.3 \times 10^7 \text{ mi}) \times (1.60934 \text{ km/mi}) = 1.49868 \times 10^8 \text{ km} \]

This process requires multiplication of the given number of miles by 1.60934 to achieve the equivalent distance in kilometers. Mastering this conversion ensures seamless data interpretation and application across various contexts, such as international navigation and scientific research.

Kilometers to Centimeters Conversion

Converting kilometers to centimeters is usually the next step when more precision is required in scientific calculations. Since 1 kilometer equals 100,000 centimeters, this conversion is straightforward but crucial for accurate smaller-scale measurements.

To convert a distance from kilometers to centimeters, the equation is:
\[ (1.49868 \times 10^8 \text{ km}) \times (100,000 \text{ cm/km}) = 1.49868 \times 10^{13} \text{ cm} \]

You simply multiply the number of kilometers by 100,000. This conversion results in an exponentially larger number, characteristic of distances measured in smaller units such as centimeters. This process is key for experiments and calculations where precise measurements are imperative, such as in engineering and physics.