Question

(a) The diameter of Earth at the equator is \(12756.27 \mathrm{~km}\). Round this number to three significant figures and express it in standard exponential notation. (b) The circumference of Earth through the poles is \(40,008 \mathrm{~km}\). Round this number to four significant figures and express it in standard exponential notation.

Short answer

(a) The diameter of Earth at the equator, rounded to three significant figures and expressed in standard exponential notation, is \(1.27\times 10^4~\mathrm{km}\). (b) The circumference of Earth through the poles, rounded to four significant figures and expressed in standard exponential notation, is \(4.001\times 10^4~\mathrm{km}\).

Step-by-step solution

Round the diameter of Earth at the equator to three significant figures.

To round the given number \(12756.27\mathrm{~km}\) to three significant figures, we should identify the third significant digit, which is 7. Since the next digit is greater than 5, we need to round up the third significant digit. The rounded number is \(12700\mathrm{~km}\).

Express the rounded number in standard exponential notation.

The standard exponential notation of a number is in the form \(A\times 10^B\), where \(A\) has only one non-zero digit to the left of the decimal point and \(B\) is an integer. To convert \(12700\mathrm{~km}\) to this notation, we can rewrite it as \(1.27\times 10^4~\mathrm{km}\). Solution for (b):

Round the circumference of Earth through the poles to four significant figures.

To round the given number \(40008\mathrm{~km}\) to four significant figures, we should identify the fourth significant digit, which is 0. Since the next digit is less than 5, we can keep the fourth significant digit as it is. The rounded number is \(40010\mathrm{~km}\).

Express the rounded number in standard exponential notation.

To convert \(40010\mathrm{~km}\) to the standard exponential notation, we can rewrite it as \(4.001\times 10^4~\mathrm{km}\). Thus, (a) The diameter of Earth at the equator, rounded to three significant figures and expressed in standard exponential notation, is \(1.27\times 10^4~\mathrm{km}\). (b) The circumference of Earth through the poles, rounded to four significant figures and expressed in standard exponential notation, is \(4.001\times 10^4~\mathrm{km}\).

Key concepts

Exponential Notation

Exponential notation is a way of writing numbers that accommodates values that are either extremely large or small. This makes them easier to work with, especially in scientific calculations. In exponential notation, a number is expressed as the product of two factors:

• A number that is usually between 1 and 10 (excluding 10), known as the coefficient.
• Ten raised to an exponent, which indicates how many places the decimal point has been moved.

When we talk about expressing the diameter of Earth in exponential notation, we mean transforming the rounded number 12700 km to a form like 1.27 × 10⁴ km.
This indicates that the decimal has moved four places to the left to place only one non-zero digit to the left of the decimal.
Exponential notation is crucial for maintaining significant figures in science, thereby ensuring that the precision of measurements is preserved in calculations.

Rounding Numbers

Rounding numbers involves reducing the number of significant digits while still keeping the value approximately the same.
It's particularly useful when you want to make numbers easier to work with or report. Here’s how you can do it:

• Identify the digit to which you're rounding. This could depend on how precise you need your number to be.
• Look at the digit immediately to the right. If it's 5 or higher, increase the rounding digit by one.
• If it's less than 5, keep the rounding digit the same and eliminate all digits to the right.

For instance, when rounding the Earth's diameter of 12756.27 km, we focus on three significant figures and note that the third figure, 7, needs to be rounded up because the next digit, 5, is greater than five,
resulting in a rounded number of 12700 km.
Remember, rounded numbers should always reflect the precision required by the context of their use.

Measurement Precision

Measurement precision is all about how detailed and exact a measurement is. This is pivotal in sciences where exact data collection and representation can affect results and interpretations.
Precision doesn’t necessarily equate to accuracy, but consistently getting the same measurement increases reliability.

• Precision in measurements is often indicated by the number of significant figures used. More significant figures mean higher precision.
• In calculations, it is crucial to carry through precision in intermediate steps and round only at the final calculation step.
• Be mindful that precision comes with limitations; the precision of your result cannot exceed the least precise measurement.

For the Earth's circumference example, rounding and precision are demonstrated by taking 40008 km and determining it to four significant figures as 40010 km.
This tells us the measurement's reliability is anchored around this level of exactness.