Question

How many minutes does it take light from the sun to reach Earth? (The distance from the sun to Earth is 93 million mi; the speed of light is \(2.99792458 \times 10^{8} \mathrm{~m} / \mathrm{s}\).)

Short answer

It takes light approximately 8.32 minutes to reach Earth from the sun.

Step-by-step solution

Convert Distance to Meters

The first step is to convert the distance from the sun to Earth from miles to meters since the speed of light is given in meters per second. To convert miles to meters, use the conversion factor: \[1 \text{ mile} = 1609.34 \text{ meters}\] Therefore, the distance in meters is: \[93,000,000 \text{ mi} \times 1609.34 \text{ m/mi} = 1.495691 \times 10^{11} \text{ m}\]

Calculate Time in Seconds

Next, calculate the time it takes for light to travel the given distance using the formula: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] Substitute the known values into the equation: \[ \text{Time} = \frac{1.495691 \times 10^{11} \text{ m}}{2.99792458 \times 10^8 \text{ m/s}} = 499.0047 \text{ seconds} \]

Convert Seconds to Minutes

Convert the time from seconds to minutes. There are 60 seconds in a minute, so divide the time in seconds by 60: \[499.0047 \text{ seconds} \div 60 = 8.316745 \text{ minutes} \]

Round to Appropriate Precision

Finally, round the answer to a reasonable degree of precision. In this context, rounding to two decimal places is typical, so the time is approximately \[8.32 \text{ minutes}\].

Key concepts

Distance Conversion

The journey of light from the sun to Earth starts with the distance between them. The problem provides this distance in miles, specifically 93 million miles. Since scientific calculations often use the metric system, we need to convert miles into meters. This is crucial because the speed of light is given in meters per second.

To do this, we use the conversion factor:

• 1 mile = 1609.34 meters

By multiplying, we convert the entire distance:

• 93,000,000 miles × 1609.34 meters/mile = 1.495691 × 10^{11} meters

This step transforms our measurement into a compatible format for further calculations. Converting units correctly ensures accuracy and helps maintain consistency within scientific equations.

Time Calculation

After aligning our distance with the speed of light in meters, we calculate the time it takes light to travel to Earth. The rule here is simple: divide the distance by the speed. Mathematics tells us:

• \[\text{Time} = \frac{\text{Distance in meters}}{\text{Speed of light in meters per second}}\]

Applying the numbers:

• \[\text{Time} = \frac{1.495691 \times 10^{11} \text{ m}}{2.99792458 \times 10^{8} \text{ m/s}} \approx 499.0047 \text{ seconds}\]

This formula strips down the problem to its essentials, using division to convert distance traversed by light into time taken. This time is, initially, in seconds—a foundation before adjusting our answer to more familiar units, like minutes.

Scientific Notation

Scientific notation is a compact way of expressing large numbers. This form is valuable in scientific calculations, like those involving the speed of light or astronomical distances. The standard format is:

• \[a \times 10^n\] where \(1 \leq a < 10\) and \(n\) is an integer

Using scientific notation:

• The distance 93 million miles becomes \(1.495691 \times 10^{11}\) meters.
• The speed of light \(2.99792458 \times 10^8\) meters per second.

This method streamlines calculations, helps track significant figures, and measures accuracy by emphasizing the most important digits first. Understanding and using scientific notation allows for easier manipulation of complex mathematical operations in science and engineering.