Problem 48 - Chemistry The Central Science In Si Units

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.

Step-by-Step Solution

Answer

(a) \(1.28 \times 10^4\) km; (b) \(4.001 \times 10^4\) km.

Step 1: Identify Significant Figures for Diameter

The diameter of Earth at the equator is given as \(12756.27 \mathrm{~km}\). Three significant figures are required. To round to three significant figures, we look at the first three digits from the left: 127.

Step 2: Rounding Off Diameter

The next digit after 127 is 5. Since this is 5 or higher, we round up the third digit (7) to 8, giving us 128.

Step 3: Express Diameter in Scientific Notation

The rounded diameter is 12800, since the other digits become zeros after rounding. In standard exponential notation, this is expressed as \(1.28 \times 10^4\) km.

Step 4: Identify Significant Figures for Circumference

The circumference of Earth through the poles is given as \(40,008 \mathrm{~km}\). Four significant figures are needed. The first four digits to focus on are 4000.

Step 5: Rounding Off Circumference

The digit after 4000 is 8, which means we must round up the last digit considered (0) to 1, giving us 40010.

Step 6: Express Circumference in Scientific Notation

The rounded circumference (40010) is expressed in standard exponential notation as \(4.001 \times 10^4\) km.

Key Concepts

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a more concise and manageable form. It's especially useful in scientific fields where such numbers are common. Instead of writing out all the zeros, scientific notation uses powers of ten to simplify numbers:

• The number is broken down into two components: a coefficient (a number between 1 and 10) and a power of ten.
• For example, the number 12800 can be expressed as \(1.28 \times 10^4\). Here, 1.28 is the coefficient, and 10 raised to the power of 4 represents the four digits that follow.
• This method not only streamlines data, but it also makes calculations and comparisons much faster and simpler.

When you encounter large figures like those involved in measuring the Earth, scientific notation helps present them succinctly.

Rounding Numbers

Rounding numbers is a fundamental skill that simplifies numbers while maintaining significant figures. It is crucial when exact numbers are not necessary, thereby reducing the complexity:

• For consistency, significant figures determine how precise a measurement is. Meaning a figure with three significant figures will be less precise than one with four.
• When rounding, assess the digit immediately following your desired level of precision. If it's 5 or greater, increase the final digit at your precision level by one. Otherwise, maintain the number.
• In the exercise, the diameter of the Earth was rounded from 12756.27 to 12800 by evaluating the fourth digit, which was 5, necessitating a rounding up of the third digit.

This method preserves the integrity of important figures while supplying a simpler number for calculations or reporting.

Earth's Measurements

When studying planetary measures such as those of Earth's, numbers are often large and unwieldy. Understanding these figures involves some geographic knowledge and mathematics:

• The Earth's equatorial diameter, for instance, is the distance across the Earth at the equator. This number is 12756.27 km before rounding.
• Similarly, the circumference of Earth through the poles measures the distance around the Earth from pole to pole, listed as 40008 km before adjustments.
• Given the immense size of Earth, these numbers help scientists and geographers understand Earth's shape and physical characteristics.

Both these measurements are typically presented in kilometers and are indispensable for calculations regarding Earth's mass, volume, and curvature.