Problem 48 - Chemistry The Central Science

Question

(a) The diameter of Earth at the equator is 7926.381 \(\mathrm{mi}\) . 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

The rounded diameter of Earth at the equator is \(7.93 \times 10^3 \ \mathrm{mi}\) and the rounded circumference of Earth through the poles is \(4.001 \times 10^4 \ \mathrm{km}\).

Step 1: Rounding the number to three significant figures

To round the number 7926.381 to three significant figures, we can follow these steps: 1. Identify the first three digits starting from the left: 7, 9, and 2. 2. Check the fourth digit: 6 3. If the fourth digit is greater than or equal to 5, add 1 to the third digit (2 + 1 = 3) 4. The rounded number will have only the first three digits and will replace the rest with zeroes: 7930 mi

Step 2: Expressing the rounded number in standard exponential notation

To express the rounded number 7930 mi in standard exponential notation, we can follow these steps: 1. Identify the position of the decimal point: In this case, it is at the end of the number (after the zero). 2. Move the decimal point to the right of the first digit (between 7 and 9). 3. Count the number of times the decimal point was moved (3 positions to the left). 4. Write the number as the product of the number without the leading zeros and a power of 10 (10 raised to the number of positions the decimal point was moved). 5. The number in standard exponential notation is: \(7.93 \times 10^3 \ \mathrm{mi}\) # Part (b) # For part (b), we are given the circumference of Earth through the poles as 40,008 km, and we have to round it to four significant figures and express it in standard exponential notation.

Step 1: Rounding the number to four significant figures

To round the number 40,008 to four significant figures, we can follow these steps: 1. Identify the first four digits starting from the left: 4, 0, 0, and 0. 2. Check the fifth digit: 8 3. If the fifth digit is greater than or equal to 5, add 1 to the fourth digit (0 + 1 = 1) 4. The rounded number will have only the first four digits and will replace the rest with zeroes: 40,010 km

Step 2: Expressing the rounded number in standard exponential notation

To express the rounded number 40,010 km in standard exponential notation, we can follow these steps: 1. Identify the position of the decimal point: In this case, it is at the end of the number (after the last zero). 2. Move the decimal point to the right of the first non-zero digit (between 4 and 0). 3. Count the number of times the decimal point was moved (4 positions to the left). 4. Write the number as the product of the number without the leading zeros and a power of 10 (10 raised to the number of positions the decimal point was moved). 5. The number in standard exponential notation is: \(4.001 \times 10^4 \ \mathrm{km}\)

Key Concepts

Standard Exponential Notation

When dealing with very large or very small numbers, it's often useful to express them in a concise form known as standard exponential notation, also frequently referred to as scientific notation. This method simplifies numbers by transforming them into a product of a number and a power of ten. The number before the multiplication sign, known as the coefficient, should be greater than or equal to 1 but less than 10.

For example, in the diameter of Earth at the equator, we rounded 7926.381 miles to 7930 miles and then expressed it in standard exponential notation as \(7.93 \times 10^3\ miles\). Here, the decimal point was moved three places to the left, which gave us the exponent of three in \(10^3\). This not only makes the number easier to read, but also easier to work with in calculations, particularly when these numbers are used in multiplication or division.

Rounding Numbers

Rounding numbers is a basic yet critical skill in chemistry and other sciences, to ensure that the precision of numbers reflects the limitations of the measurement tools used to acquire them. To round a number to a specific number of significant figures, follow these guidelines: Identify the digits that will remain after rounding, then look at the subsequent digit. If it is 5 or greater, increase the last remaining digit by one; if it is less than 5, leave the last digit as is.

For instance, the number 7926.381 was rounded to three significant figures, which resulted in 7930 after applying the process of rounding. The fourth digit, '6', was greater than 5, so we increased the third digit, '2', to '3'. This rounding process ensures that your numbers reflect both the precision of the measurement and the level of detail you require for calculations or comparisons.

Scientific Notation

Scientific notation is a form of writing numbers that are too big or too small to be conveniently written in decimal form. It is similar to standard exponential notation and is widely used in science to handle the wide range of values encountered—from the mass of a proton to the distance between galaxies. A number is written in scientific notation as the product of a number between 1 and 10 and a power of ten.

The circumference of Earth through the poles was given as 40,008 km and rounded to 40,010 km to four significant figures. In scientific notation, it's written as \(4.001 \times 10^4\ km\). Remember, the power of 10 reflects the number of places the decimal point has moved from its original position. Scientific notation is ideal for simplifying numbers and making them easier to understand, compare, and use in further computations.