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Chapter 1: Problem 48

(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.

Short Answer

Expert verified
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 by step solution

01

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
02

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.
03

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
04

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: \(4.001 \times 10^4 \ \mathrm{km}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攆rom 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.

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Most popular questions from this chapter

(a) To identify a liquid substance, a student determined its density. Using a graduated cylinder, she measured out a 45 -mL. sample of the substance. She then measured the mass of the sample, finding that it weighed 38.5 \(\mathrm{g}\) . She knew that the substance had to be either isopropylalcohol (density 0.785 \(\mathrm{g} / \mathrm{mL}\) )or toluene (density 0.866 \(\mathrm{g} / \mathrm{mL} ) .\) What are the calculated density and the probable identity of the substance? (b) An experiment requires 45.0 \(\mathrm{g}\) of ethylene glycol, a liquid whose density is 1.114 \(\mathrm{g} / \mathrm{mL}\) . Rather than weigh the sample on a balance, a chemist chooses to dispense the liquid using a graduated cylin-der. What volume of the liquid should he use? (c) Is a graduated cylinder such as that shown in Figure 1.21 likely to afford the accuracy of measurement needed? (d) A cubic piece of metal measures 5.00 \(\mathrm{cm}\) on each edge. If the metal is nickel, whose density is \(8.90 \mathrm{g} / \mathrm{cm}^{3},\) what is the mass of the cube?

(a) A cube of osmium metal 1.500 \(\mathrm{cm}\) on a side has a mass of 76.31 \(\mathrm{g}\) at \(25^{\circ} \mathrm{C}\) . What is its density in \(\mathrm{g} / \mathrm{cm}^{3}\) at this temperature? (b) The density of titanium metal is 4.51 \(\mathrm{g} / \mathrm{cm}^{3}\) at \(25^{\circ} \mathrm{C}\) . What mass of titanium displaces 125.0 \(\mathrm{mL}\) of water at \(25^{\circ} \mathrm{C} ?\) (c) The density of benzene at \(15^{\circ} \mathrm{C}\) is 0.8787 \(\mathrm{g} / \mathrm{mL}\) . Calculate the mass of 0.1500 L of benzene at this temperature.

Use appropriate metric prefixes to write the following measurements without use of exponents: (a) \(2.3 \times 10^{-10} \mathrm{L}\) ,(b) \(4.7 \times 10^{-6} \mathrm{g},\) (c) \(1.85 \times 10^{-12} \mathrm{m},\) (d) \(16.7 \times 10^{6} \mathrm{s}\) (e) \(15.7 \times 10^{3} \mathrm{g},(\mathrm{f}) 1.34 \times 10^{-3} \mathrm{m},(\mathrm{g}) 1.84 \times 10^{2} \mathrm{cm}\)

In the United States, water used for irrigation is measured in acre-feet. An acre-foot of water covers an acre to a depth of exactly 1 ft. An acre is 4840 yd. An acre-foot is enough water to supply two typical households for 1.00 yr. (a) If desalinated water costs \(\$ 1950\) per acre-foot, how much does desalinated water cost per liter? (b) How much would it cost one household per day if it were the only source of water?

Silicon for computer chips is grown in large cylinders called aboules" that are 300 \(\mathrm{mm}\) in diameter and 2 \(\mathrm{m}\) in length, as shown. The density of silicon is 2.33 \(\mathrm{g} / \mathrm{cm}^{3} .\) Silicon wafers for making integrated circuits are sliced from a 2.0 -m boule and are typically 0.75 \(\mathrm{mm}\) thick and 300 \(\mathrm{mm}\) in diameter.(a) How many wafers can be cut from a single boule? (b) What is the mass of a silicon wafer? (The volume of a cylinder is given by \(\pi r^{2} h,\) where \(r\) is the radius and \(h\) is its height.)
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