Large & Small Numbers

Transcription

1 Large & Small Numbers Scientists frequently work with very large or small numbers. Astronomers work with galaxies that contain billions of stars at great distances from us. On the other hand, biologists work with cells that can only be seen under the microscope. These cells are made of atoms that are too small to be magnified sufficiently to be seen under a microscope. It therefore becomes necessary to express these numbers conveniently and to use methods of approximating them. SCIENTIFIC NOTATION In scientific notion a number is expressed in the form M x 10 e where (M) is a number less than 10.0 but equal to or greater than 1.0 (called the mantissa), and (e) is a whole number integer (called the exponent). The exponent represents the placement of the decimal. Numbers greater than one have a positive exponent ( 3.4 x 10 2 )and numbers less than one and greater than zero have negative exponents ( 6.2 x 10-1 ). Both large and small numbers can be expressed as numbers multiplied by 10, 100, 1000, 1/10, 1/100, 1/1000, etc, each of which can then be rewritten as a power of 10. Study the following examples = 5.2 x 1000 = 5.2 x 10 x 10 x 10 = 5.2 x = 5.2 x 100 = 5.2 x 10 x 10 = 5.2 x = 5.2 x = 5.2 x 1/1000 = 5.2 x 1/10 x 1/10 x 1/10 = 5.2 x = 5.2 x 0.01 = 5.2 x 1/100 = 5.2 x 1/10 x 1/10 = 5.2 x10-2 PRACTICE PROBLEMS Express the following numbers in scientific notation x x x x 10 3 Express each of the following numbers to ordinary notation x x x x MEASURING LENGTH When making measurements, it must be remembered that the values recorded represent the accuracy of the instrument used in making these measurements. In the previous laboratory, when using the triple beam balance the masses of objects were recorded to the nearest 0.01 g, when using the 100 ml graduated cylinder volumes were recorded to the nearest 0.1 ml. You are already familiar with making measurements of length using the English system, such as inches, feet, yards, and miles. The metric system uses units of millimeters, centimeters, meters, and kilometers for determining length. Examine the ruler. The smallest graduations on the metric ruler are millimeters, while the larger graduations are marked in centimeters. Notice that there are approximately 2.5 centimeters per inch (2.54 to be exact) or 25 millimeters per inch (25.4 to be exact.) Making measurements using the ruler are similar to the graduated cylinder and balance. Due to the small spaces between millimeter graduations, the smallest estimations that are possible on the ruler are 0.1 mm, or 0.01 cm. Large & Small Numbers 1

2 PRACTICE Use the ruler and measure the length of the line below. Record the measurement in millimeters, centimeters and meters. Have your instructor check your measurements mm cm m Turn to Check Your Understanding at the end of this lab and record the lengths given on the rulers. SIGNIFICANT FIGURES The number of figures expressed in calculations must represent the number of figures used in making these measurements. As a rule, calculations obtained from measurements cannot have more significant figures than the least accurate number. For example, in measuring the area of a sheet of paper, the following measurements were obtained: Length = cm Width = 9.54 cm The area of the paper is obtained by multiplying the length and width of the paper: Area = cm x 9.54 cm = cm 2 Using the rule stated above, the least accurate number contains 3 significant figures, and the area of the sheet of paper must be recorded as 242. cm 2. Notice that the area was rounded up since the digits after the last significant figure were greater than 0.5. PART I: MEASURING LARGE OBJECTS Measure the sides of the rectangular solid, and express these measurements in millimeters, centimeters, and meters. From these measurements determine the volume of the block expressing the appropriate significant figures and in scientific notation. Millimeters Centimeters Meters Volume mm 3 cm 3 m x x x 10-4 Large & Small Numbers 2

3 PART II: MEASURING SMALL OBJECTS BEANS IN A VIAL The general procedure you will use in this determination is to (1) estimate the volume of an average bean, (2) calculate the total volume of the vial, and (3) determine the number of beans in the vial by dividing the total volume of the vial by the average volume of a bean. PROCEDURE 1. The Volume of a Bean Select TWO representative beans from the vial. Determine their volume by assuming that they are rectangular solids. Measurements and Calculations: Bean Sample 1 2 AVERAGE VOLUME: 1100 mm 3 or 1.1 cm mm mm 3 = / 2 = mm 3 ( 1100 mm 3 ) (2 significant figures) cm mm 3 = / 2 = cm 3 (1.1 cm 3 ) (2 significant figures) 2. The Volume of the Vial The vial is cylindrical in shape. The volume of any regular shaped object can be obtained by multiplying the area of the base by the height. Record the measurements of the vial and the volume calculations below using appropriate significant figures. Measurements and Calculations: Height Width Length Volume 9.2 mm 0.92 cm 8.7 mm 0.87 cm 12.4 mm 1.24 cm 13.2 mm 1.32 cm 8.5 mm 0.85 cm 10.0 mm 1.00 cm (Volume of a cylinder = π x radius 2 x height) Diameter Height Volume 43.0 mm mm 150, mm cm cm cm mm cm mm cm 3 VOLUME OF THE VIAL: 151,000 mm 3 or 151 cm 3 (3 significant figures) 3. Number of Beans in the Vial If the volume of the vial is divided by the average volume of the beans, the number of beans in the vial is obtained. Calculate the number of beans in the vial below using appropriate significant figures and in scientific notation. 151,000 mm 3 / 1100 mm 3 = or 140 beans 1.4 x 10 2 beans 151 cm 3 / 1.1 cm 3 = or 140 beans CALCULATED NUMBER OF BEANS: 140 (Note: The measurement for the bean has 2 significant figures, therefore the calculated number of beans in the vial must be represented by 2 significant figures.) Open the vial and count the number of beans within the vial. ACTUAL NUMBER OF BEANS: 127 Large & Small Numbers 3

4 PART III: MEASURING SMALLER OBJECTS: GRAINS OF SALT IN A VIAL You have a vial partially filled with salt. Using the same procedures used for the beans, determine the number of grains of salt in the vial. In the space below, construct a data table, and show any calculations that you used. (Hint: You can determine the dimensions of salt granules (cubes) by looking at them under a hand lens. Determine their size by measuring 10 salt granules arranged end-to-end viewed under the lens.) Construct a data table to include the information you collected. Length of 10 salt granules Length of 1 salt granule Diameter of vial Height of salt granules in vial 0.45 cm or 4.5 mm cm or 0.45 mm 0.90 cm or 9.0 mm 3.20 cm or 32.0 mm Calculate the volume of one salt granule. Express your answer in scientific notation and with appropriate significant figures. Volume = width x length x height (0.45 mm) x (0.45 mm) x (0.45 mm) = mm3 = 9.1 x 10-2 mm 3 (2 significant figures) (0.045 cm) x (0.045 cm) x (0.045 cm) = cm3 = 9.1 x 10-5 cm 3 Calculate the number of salt granules in the vial. Show your calculations and express your answer in scientific notation using appropriate significant figures. Volume of vial = π x radius 2 x height (3.14) x (4.5 mm) 2 x (32.0 mm) = 2, mm 3 = 2,000 mm 3 = 2.0 x 10 3 mm 3 (2 significant figures) (3.14) x (0.45 cm) 2 x (3.20 cm) = cm 3 = 2.0 x 10 1 cm 3 (2 significant figures) # salt granules = volume of vial / volume of 1 granule 2.0 cm 3 / 9.1 x 10-5 cm 3 = 21, = 22,000 = 2.2 x 10 4 granules (2 significant figures) 2.0 x 10 3 mm 3 / 9.1 x 10-2 mm 3 = 21, = 22,000 = 2.2 x 10-4 granules CALCULATED NUMBER OF SALT GRANULES: 22,000 or 2.2 x 10 4 Large & Small Numbers 4

5 POSTLAB QUESTIONS 1. Suppose that an ice cube measures 1.0 cm on an edge. Calculate the number of ice cubes that could be made from a cubic iceberg which measures 1.0 kilometer on an edge. Express your answer using correct significant figures and in scientific notation. (Hint: Determine the volume of the iceberg in cubic centimeters.) 1 km = 1000 m = 100,000 cm Volume = width x length x height Volume iceberg = (100,000 cm) 3 = (1.0 x 10 5 cm) 3 = 1.0 x cm 3 1 km 1 km 1 km # ice cubes = volume iceberg / volume ice cube = 1.0 x cm 3 / 1.0 cm 3 = 1.0 x ice cubes 2. A vial which has a diameter of cm and a height of cm is filled with grains of rice which measure 1.0 mm x 1.0 mm x 4.8 mm. Estimate the number of grains of rice in this vial using correct significant digits and in scientific notation. Show your calculations below. Note: The units given for the vial and the rice are not in the same units. You should convert one set of data to match the other before beginning the calculations. Volume of one grain of rice: 4.8 mm 3 or cm 3 (2 significant figures) Volume of vial: mm 3 or 2028 cm 3 (4 significant figures) Grains of rice: 42,000 or 4.2 x 10 4 grains Large & Small Numbers 5

6 Check Your Understanding millimeters centimeters meters , , Large & Small Numbers 6