Solve Problems Using Scientific Notation

Transcription

1 Domain 2 Lesson 8 Solve Problems Using Scientific Notation Common Core Standards: 8.EE.3, 8.EE.4 Getting the Idea Sometimes, you may need to multiply or divide numbers written in scientific notation in order to solve real-world problems. Example 1 A rectangular section of wilderness will be set aside as a new wildlife refuge. Its dimensions are meters by meters. Find the area of the land in square meters. Then convert the area into square kilometers using the conversion below. 1 square kilometer ( km 2 ) square meters ( m 2 ) Which unit is a better choice for measuring the area of the wildlife refuge, and why? Strategy Multiply the dimensions. Convert the area into square kilometers. Compare the two units. Step 1 Multiply the dimensions to find the area, A. A (in m 2 ) 5 ( m)( m) Step 2 Step 3 Step 4 Multiply the first factors and then multiply the power-of-10 factors So, A (in m 2 ) 5 ( m)( m) Rewrite the number in scientific notation Convert the area into square kilometers. To convert a smaller unit (square meters) to a larger unit (square kilometers), divide: A (in km 2 ) Divide the first factors and then divide the power-of-10 factors So, A (in km 2 )

2 Step 5 Solution Which is the better unit to use? m ,000,000,000 m km ,000 km 2 20,000 is a more reasonable number to work with in standard form. Also, square kilometers are larger units than square meters. Since the area is large, it is better to use the larger unit. The area of the refuge is square meters or square kilometers. Square kilometers is a better unit to use because the area is very large. Sometimes, you may use technology, such as a calculator, to generate a number. If the result is a number that is very large or very small, many calculators will automatically give the number in scientific notation. Example 2 One cubic millimeter of Ms. Murphy s blood contains about 5,000,000 red blood cells. There are about 4,900,000 cubic millimeters of blood in her entire body. Use a calculator to determine approximately how many red blood cells Ms. Murphy has in total. Interpret the number your calculator gives as the final answer. Strategy Step 1 Step 2 Use a calculator to determine the answer. Interpret the result. How can you find the total number of red blood cells? Multiply the number of red blood cells in one cubic millimeter of blood (5,000,000) by the total number of cubic millimeters of blood in the body (4,900,000). Use a calculator to determine the answer. Type Press Step 3 Solution Type Press Interpret the answer shown on the calculator display. The screen shows this: Ms. Murphy has a total of about red blood cells in her body. 77

3 Example 3 California, the most populous state, has approximately people living in it. The population of the entire United States is approximately people. About how many times greater is the population of the United States than the population of California? Strategy Step 1 Step 2 Solution Decide if you should multiply or divide. Then solve the problem. Decide on which operation to use. To find how many times greater, divide by Divide the first factors and then divide the power-of-10 factors So, The population of the United States is 7 1 times the population of 2 California. Coached Example A computer was used to draw a rectangle with an area of square meter. Would it be better to measure the area in square meters or square millimeters? Use the conversion below to help determine your answer. 1 square meter ( m 2 ) square millimeters ( mm 2 ) Rewrite in scientific notation The decimal point was moved places to the right. The original number is less than, so the exponent will be negative Multiply to convert that number of square meters to square millimeters: ( )( ) Multiply the first factors: Multiply the power-of-10 factors: The area is square millimeters. It is better to measure the area in square because it is better to measure a small area using a unit. 78 Domain 2: Expressions and Equations

4 Lesson 8: Solve Problems Using Scientific Notation Lesson Practice Choose the correct answer. 1. One microgram is equal to gram. If the mass of a substance is micrograms, what is its mass in grams? A gram B gram C grams D grams 2. A rectangular section of land made up of wheat farms has a length of meters and a width of meters. What is the area of the land in square meters? A square meters B square meters C square meters D square meters 3. A microscope is set so it makes an object appear times larger than its actual size. A virus has a diameter of meter. How large will the diameter of the virus appear when it is viewed under the microscope? A meter B meter C meter D meters 4. Neptune is approximately kilometers in diameter. Mars is approximately kilometers in diameter. Which is an accurate comparison of the diameters of these two planets? A. The diameter of Neptune is more than 7 times greater than the diameter of Mars. B. The diameter of Mars is more than 7 times greater than the diameter of Neptune. C. The diameter of Neptune is about 1.4 times greater than the diameter of Mars. D. The diameter of Mars is about 1.4 times greater than the diameter of Neptune. 5. A box contains paper clips. The mass of each paper clip in the box is kilogram. What is the combined mass of the paper clips in the box? A. 4 kilograms B. 40 kilograms C kilograms D kilogram 6. The head of a pin has a diameter of meter. A bacterium has a diameter of meter. How many bacteria that size would fit across the diameter of the pinhead? A. 2 C. 200 B. 20 D

5 7. The population of Canada is approximately The population of Mexico is approximately Which statement accurately compares the populations of Canada and Mexico? A. The population of Canada is more than 30 times greater than the population of Mexico. B. The population of Mexico is more than 30 times greater than the population of Canada. C. The population of Canada is more than 3 times greater than the population of Mexico. D. The population of Mexico is more than 3 times greater than the population of Canada. 8. One nanometer is equivalent to meters. Which is equivalent to 0.3 nanometers? A meters B meters C meter D meter 9. A rectangular yard has a length of kilometer and a width of kilometer. A. Use scientific notation to express the area of the yard in square kilometers, showing each step in the process. B. Convert the area into square meters using the conversion below. 1 square kilometer ( km 2 ) square meters ( m 2 ) Give your answer in standard form. Which unit is a better choice for measuring the area of the yard, and why? 80 Domain 2: Expressions and Equations

6 Lesson 8: Solve Problems Using Scientific Notation 10. An island has an area of approximately square miles. The island has a population of about Circle the number in scientific notation that best completes the statement The number of people per square mile on the island is about A national forest is in the shape of a square with sides measuring miles. Circle the number in scientific notation that best completes the statement The area, in square miles, of the national forest is Draw a line from each word problem to its solution. A. Lake Superior has roughly 25 times the volume of Lake Erie. The volume of Lake Superior is approximately cubic kilometers. About what is the volume, in cubic kilometers, of Lake Erie? B. A pint is equivalent to about cubic millimeters. There are roughly red blood cells in 1 cubic millimeter of human blood. About how many red blood cells are in a pint of human blood? C. The area of the Atlantic Ocean is about square miles. The area of the Indian Ocean is roughly 7 that of the 10 Atlantic Ocean. About what is the area, in square miles, of the Indian Ocean?

7 13. Look at each word problem. Is the solution? Select Yes or No. A. The speed of light is meters per second. If the sun is meters from Earth, how many seconds does it take light to reach Earth? Yes No B. A rectangular field has a length of miles and a width of miles. What is the area of the field in square miles? C. Isa had dollars in her savings account. She withdrew _ 1 of the money from her savings 5 account. How much money, in dollars, is left in her savings account? Yes No Yes No 14. Circle every word problem that has a solution of A. A microscope is set so that an object looks times as large as its actual size. A virus has a radius of meters. What is the measure, in meters, of the radius of the virus when seen under the microscope? B. The surface area of a paper clip is Jorge painted three paper clips for an art project. What is the total surface area he painted? C. The head of a pin has a diameter of meters. A bacterium has a diameter of meters. How many bacteria can fit on the pin head? 15. Alpha Centauri, the star nearest to Earth, is 4.3 light years from Earth. A light year is miles. Circle the number in scientific notation that best completes the statement. The distance from Earth to Alpha Centauri is miles. 82 Domain 2: Expressions and Equations