{"id":15340,"date":"2021-10-11T23:05:33","date_gmt":"2021-10-11T23:05:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/4-1-4-scientific-notation\/"},"modified":"2021-10-16T20:34:20","modified_gmt":"2021-10-16T20:34:20","slug":"4-1-4-scientific-notation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/4-1-4-scientific-notation\/","title":{"raw":"Problem Solving With Scientific Notation","rendered":"Problem Solving With Scientific Notation"},"content":{"raw":"\n<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Solve application problems involving scientific notation<\/li>\n<\/ul>\n<\/div>\n&nbsp;\n\n[caption id=\"attachment_4384\" align=\"aligncenter\" width=\"361\"]<img class=\"wp-image-4384\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/26225033\/Screen-Shot-2016-05-26-at-3.50.25-PM-300x197.png\" alt=\"Molecule of water with one oxygen bonded to two hydrogen.\" width=\"361\" height=\"237\"> Water Molecule[\/caption]\n<h2 id=\"title3\">Solve application problems<\/h2>\nLearning rules for exponents&nbsp;seems pointless without context, so let's explore some examples of using scientific notation that involve real problems. First, let's look at an example of how scientific notation can be used to describe real measurements.\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\nMatch each length in the table with the appropriate number of meters described in scientific notation below.\n<table>\n<tbody>\n<tr>\n<td>The height of a desk<\/td>\n<td>Diameter of&nbsp;water molecule<\/td>\n<td>Diameter of Sun at its equator<\/td>\n<\/tr>\n<tr>\n<td>Distance from Earth to Neptune<\/td>\n<td>Diameter of Earth at the Equator<\/td>\n<td>Height of Mt. Everest (rounded)<\/td>\n<\/tr>\n<tr>\n<td>Diameter&nbsp;of average human cell<\/td>\n<td>Diameter of a large grain of sand<\/td>\n<td>Distance a bullet travels in one second<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<thead>\n<tr>\n<td>Power of [latex]10[\/latex], units in meters<\/td>\n<td>Length from table above<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]10^{12}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{9}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{6}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{4}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{2}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{0}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-3}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-5}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-10}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[reveal-answer q=\"993302\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"993302\"]\n<table>\n<thead>\n<tr>\n<td>Power of [latex]10[\/latex], units in meters<\/td>\n<td>Length from table above<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]10^{12}[\/latex]<\/td>\n<td>Distance from Earth to Neptune<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{9}[\/latex]<\/td>\n<td>Diameter of Sun at it's Equator<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{6}[\/latex]<\/td>\n<td>Diameter of Earth at the Equator<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{4}[\/latex]<\/td>\n<td>Height of Mt. Everest (rounded)<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{2}[\/latex]<\/td>\n<td>Distance a bullet travels in one second<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{0}[\/latex]<\/td>\n<td>The height of a desk<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-3}[\/latex]<\/td>\n<td>Diameter of a large grain of sand<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-5}[\/latex]<\/td>\n<td>Diameter&nbsp;of average human cell<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-10}[\/latex]<\/td>\n<td>Diameter of&nbsp;water molecule<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n\n[caption id=\"attachment_4382\" align=\"alignright\" width=\"181\"]<img class=\" wp-image-4382\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/26224141\/Screen-Shot-2016-05-26-at-3.41.43-PM-300x210.png\" alt=\"Red Blood Cells.\" width=\"181\" height=\"126\"> Red Blood Cells[\/caption]\n\n&nbsp;\n\nOne of the most important parts of solving a \"real\" problem is translating the words into appropriate mathematical terms, and recognizing when a well known formula may help.&nbsp;Here's an example that requires you to find the density&nbsp;of a cell, given its mass and volume. Cells aren't visible to the naked eye, so their measurements, as described with scientific notation, involve negative exponents.\n\n&nbsp;\n\n&nbsp;\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nHuman cells come in a wide variety of shapes and sizes. The mass of an average human cell is about [latex]2\\times10^{-11}[\/latex] grams[footnote]Orders of magnitude (mass). (n.d.). Retrieved May 26, 2016, from https:\/\/en.wikipedia.org\/wiki\/Orders_of_magnitude_(mass)[\/footnote]Red blood cells are one of the smallest types of cells[footnote]How Big is a Human Cell?[\/footnote], clocking in at a volume of approximately [latex]10^{-6}\\text{ meters }^3[\/latex].[footnote]How big is a human cell? - Weizmann Institute of Science. (n.d.). Retrieved May 26, 2016, from http:\/\/www.weizmann.ac.il\/plants\/Milo\/images\/humanCellSize120116Clean.pdf[\/footnote] Biologists have recently discovered how to use the density of some types of cells to indicate the presence of disorders such as sickle cell anemia or leukemia. [footnote]Grover, W. H., Bryan, A. K., Diez-Silva, M., Suresh, S., Higgins, J. M., &amp; Manalis, S. R. (2011). Measuring single-cell density. Proceedings of the National Academy of Sciences, 108(27), 10992-10996. doi:10.1073\/pnas.1104651108[\/footnote]&nbsp;Density is calculated as the ratio of [latex]\\frac{\\text{ mass }}{\\text{ volume }}[\/latex]. Calculate the density of an average human cell.\n\n[reveal-answer q=\"856454\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"856454\"]\n\n<strong>Read and Understand:&nbsp;<\/strong>We are given an average cellular mass and volume as well as the formula for density. We are looking for the density of an average human cell.\n\n<strong>Define and Translate:&nbsp;<\/strong>&nbsp;[latex]m=\\text{mass}=2\\times10^{-11}[\/latex], [latex]v=\\text{volume}=10^{-6}\\text{ meters}^3[\/latex], [latex]\\text{density}=\\frac{\\text{ mass }}{\\text{ volume }}[\/latex]\n\n<strong>Write and Solve:&nbsp;<\/strong>Use the quotient rule to simplify the ratio.\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\text{ density }=\\frac{2\\times10^{-11}\\text{ grams }}{10^{-6}\\text{ meters }^3}\\\\\\text{ }\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=2\\times10^{-11-\\left(-6\\right)}\\frac{\\text{ grams }}{\\text{ meters }^3}\\\\\\text{ }\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=2\\times10^{-5}\\frac{\\text{ grams }}{\\text{ meters }^3}\\\\\\end{array}[\/latex]<\/p>\nIf scientists know the density of healthy cells, they can compare the density of a sick person's&nbsp;cells to that to rule out or test for disorders or diseases that may affect cellular density.\n<h4 style=\"text-align: left;\">Answer<\/h4>\n<p style=\"text-align: left;\">The average density of a human cell is [latex]2\\times10^{-5}\\frac{\\text{ grams }}{\\text{ meters }^3}[\/latex]<\/p>\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\n\n<\/div>\nThe following video provides an example of how to find the number of operations a computer can perform in a very short amount of time.\n\nhttps:\/\/youtu.be\/Cbm6ejEbu-o\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question]88501[\/ohm_question]\n\n<\/div>\n&nbsp;\n\n[caption id=\"attachment_4381\" align=\"alignright\" width=\"342\"]<img class=\" wp-image-4381\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/26223918\/Screen-Shot-2016-05-26-at-3.39.02-PM-300x137.png\" alt=\"Earth in the foreground, sun in the background, light beams traveling from the sun to the earth.\" width=\"342\" height=\"156\"> Light traveling from the sun to the earth.[\/caption]\n\n&nbsp;\n\nIn the next example, you will use another well known formula, [latex]d=r\\cdot{t}[\/latex], to find how long it takes light to travel from the sun to the earth. Unlike the previous example, the distance between the earth and the sun is massive, so the numbers you will work with have positive exponents.\n\n&nbsp;\n\n&nbsp;\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nThe speed of light is [latex]3\\times10^{8}\\frac{\\text{ meters }}{\\text{ second }}[\/latex]. If the sun is [latex]1.5\\times10^{11}[\/latex] meters from earth, how many seconds does it take for sunlight to reach the earth? &nbsp;Write your answer in scientific notation.\n[reveal-answer q=\"532092\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"532092\"]\n\n<strong>Read and Understand:&nbsp;<\/strong>We are looking for how long\u2014an amount of time. We are given a rate which has units of meters per second and a distance in meters. This is a [latex]d=r\\cdot{t}[\/latex] problem.\n\n<strong>Define and Translate:&nbsp;<\/strong>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}d=1.5\\times10^{11}\\\\r=3\\times10^{8}\\frac{\\text{ meters }}{\\text{ second }}\\\\t=\\text{ ? }\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\"><strong>Write and Solve:&nbsp;<\/strong>Substitute the values we are given into the&nbsp;[latex]d=r\\cdot{t}[\/latex] equation. We will work without units to make it easier. Often, scientists will work with units to make sure they have made correct calculations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}d=r\\cdot{t}\\\\1.5\\times10^{11}=3\\times10^{8}\\cdot{t}\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Divide both sides of the equation by&nbsp;[latex]3\\times10^{8}[\/latex] to isolate&nbsp;<em>t.<\/em><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}1.5\\times10^{11}=3\\times10^{8}\\cdot{t}\\\\\\text{ }\\\\\\frac{1.5\\times10^{11}}{3\\times10^{8}}=\\frac{3\\times10^{8}}{3\\times10^{8}}\\cdot{t}\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">On the left side, you will need to use the quotient rule of exponents to simplify, and on the right, you are left with&nbsp;<em>t.&nbsp;<\/em><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{1.5\\times10^{11}}{3\\times10^{8}}=\\frac{3\\times10^{8}}{3\\times10^{8}}\\cdot{t}\\\\\\text{ }\\\\\\left(\\frac{1.5}{3}\\right)\\times\\left(\\frac{10^{11}}{10^{8}}\\right)=t\\\\\\text{ }\\\\\\left(0.5\\right)\\times\\left(10^{11-8}\\right)=t\\\\0.5\\times10^3=t\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">This answer is not in scientific notation, so we will move the decimal to the right, which means we need to subtract one factor of [latex]10[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]0.5\\times10^3=5.0\\times10^2=t[\/latex]<\/p>\n\n<h4>Answer<\/h4>\nThe time it takes light to travel from the sun to the earth is [latex]5.0\\times10^2=t[\/latex] seconds, or in standard&nbsp;notation, [latex]500[\/latex] seconds. &nbsp;That's not bad considering how far it has to travel!\n\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n\nIn the following video&nbsp;we calculate how many miles the participants of the New York marathon ran combined, and compare that to the circumference of the earth.\nhttps:\/\/youtu.be\/15tw4-v100Y\n<h3>Summary<\/h3>\nScientific notation was developed to assist mathematicians, scientists, and others when expressing and working with very large and very small numbers. Scientific notation follows a very specific format in which a number is expressed as the product of a number greater than or equal to one and less than ten, and a power of [latex]10[\/latex]. The format is written [latex]a\\times10^{n}[\/latex], where [latex]1\\leq{a}&lt;10[\/latex]&nbsp;and <i>n <\/i>is an integer.&nbsp;To multiply or divide numbers in scientific notation, you can use the commutative and associative properties to group the exponential terms together and apply the rules of exponents.\n<h2>Contribute!<\/h2><div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div><a href=\"https:\/\/docs.google.com\/document\/d\/1sk-n1xsrRJlGvWjNjjTUYARfEfi42MnvFu8_NqY9HRI\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a>\n","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve application problems involving scientific notation<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"attachment_4384\" style=\"width: 371px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4384\" class=\"wp-image-4384\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/26225033\/Screen-Shot-2016-05-26-at-3.50.25-PM-300x197.png\" alt=\"Molecule of water with one oxygen bonded to two hydrogen.\" width=\"361\" height=\"237\" \/><\/p>\n<p id=\"caption-attachment-4384\" class=\"wp-caption-text\">Water Molecule<\/p>\n<\/div>\n<h2 id=\"title3\">Solve application problems<\/h2>\n<p>Learning rules for exponents&nbsp;seems pointless without context, so let&#8217;s explore some examples of using scientific notation that involve real problems. First, let&#8217;s look at an example of how scientific notation can be used to describe real measurements.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>Match each length in the table with the appropriate number of meters described in scientific notation below.<\/p>\n<table>\n<tbody>\n<tr>\n<td>The height of a desk<\/td>\n<td>Diameter of&nbsp;water molecule<\/td>\n<td>Diameter of Sun at its equator<\/td>\n<\/tr>\n<tr>\n<td>Distance from Earth to Neptune<\/td>\n<td>Diameter of Earth at the Equator<\/td>\n<td>Height of Mt. Everest (rounded)<\/td>\n<\/tr>\n<tr>\n<td>Diameter&nbsp;of average human cell<\/td>\n<td>Diameter of a large grain of sand<\/td>\n<td>Distance a bullet travels in one second<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<thead>\n<tr>\n<td>Power of [latex]10[\/latex], units in meters<\/td>\n<td>Length from table above<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]10^{12}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{9}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{6}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{4}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{2}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{0}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-3}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-5}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-10}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q993302\">Show Solution<\/span><\/p>\n<div id=\"q993302\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<thead>\n<tr>\n<td>Power of [latex]10[\/latex], units in meters<\/td>\n<td>Length from table above<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]10^{12}[\/latex]<\/td>\n<td>Distance from Earth to Neptune<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{9}[\/latex]<\/td>\n<td>Diameter of Sun at it&#8217;s Equator<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{6}[\/latex]<\/td>\n<td>Diameter of Earth at the Equator<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{4}[\/latex]<\/td>\n<td>Height of Mt. Everest (rounded)<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{2}[\/latex]<\/td>\n<td>Distance a bullet travels in one second<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{0}[\/latex]<\/td>\n<td>The height of a desk<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-3}[\/latex]<\/td>\n<td>Diameter of a large grain of sand<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-5}[\/latex]<\/td>\n<td>Diameter&nbsp;of average human cell<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-10}[\/latex]<\/td>\n<td>Diameter of&nbsp;water molecule<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"attachment_4382\" style=\"width: 191px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4382\" class=\"wp-image-4382\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/26224141\/Screen-Shot-2016-05-26-at-3.41.43-PM-300x210.png\" alt=\"Red Blood Cells.\" width=\"181\" height=\"126\" \/><\/p>\n<p id=\"caption-attachment-4382\" class=\"wp-caption-text\">Red Blood Cells<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>One of the most important parts of solving a &#8220;real&#8221; problem is translating the words into appropriate mathematical terms, and recognizing when a well known formula may help.&nbsp;Here&#8217;s an example that requires you to find the density&nbsp;of a cell, given its mass and volume. Cells aren&#8217;t visible to the naked eye, so their measurements, as described with scientific notation, involve negative exponents.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Human cells come in a wide variety of shapes and sizes. The mass of an average human cell is about [latex]2\\times10^{-11}[\/latex] grams<a class=\"footnote\" title=\"Orders of magnitude (mass). (n.d.). Retrieved May 26, 2016, from https:\/\/en.wikipedia.org\/wiki\/Orders_of_magnitude_(mass)\" id=\"return-footnote-15340-1\" href=\"#footnote-15340-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>Red blood cells are one of the smallest types of cells<a class=\"footnote\" title=\"How Big is a Human Cell?\" id=\"return-footnote-15340-2\" href=\"#footnote-15340-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>, clocking in at a volume of approximately [latex]10^{-6}\\text{ meters }^3[\/latex].<a class=\"footnote\" title=\"How big is a human cell? - Weizmann Institute of Science. (n.d.). Retrieved May 26, 2016, from http:\/\/www.weizmann.ac.il\/plants\/Milo\/images\/humanCellSize120116Clean.pdf\" id=\"return-footnote-15340-3\" href=\"#footnote-15340-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a> Biologists have recently discovered how to use the density of some types of cells to indicate the presence of disorders such as sickle cell anemia or leukemia. <a class=\"footnote\" title=\"Grover, W. H., Bryan, A. K., Diez-Silva, M., Suresh, S., Higgins, J. M., &amp; Manalis, S. R. (2011). Measuring single-cell density. Proceedings of the National Academy of Sciences, 108(27), 10992-10996. doi:10.1073\/pnas.1104651108\" id=\"return-footnote-15340-4\" href=\"#footnote-15340-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a>&nbsp;Density is calculated as the ratio of [latex]\\frac{\\text{ mass }}{\\text{ volume }}[\/latex]. Calculate the density of an average human cell.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q856454\">Show Solution<\/span><\/p>\n<div id=\"q856454\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Read and Understand:&nbsp;<\/strong>We are given an average cellular mass and volume as well as the formula for density. We are looking for the density of an average human cell.<\/p>\n<p><strong>Define and Translate:&nbsp;<\/strong>&nbsp;[latex]m=\\text{mass}=2\\times10^{-11}[\/latex], [latex]v=\\text{volume}=10^{-6}\\text{ meters}^3[\/latex], [latex]\\text{density}=\\frac{\\text{ mass }}{\\text{ volume }}[\/latex]<\/p>\n<p><strong>Write and Solve:&nbsp;<\/strong>Use the quotient rule to simplify the ratio.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\text{ density }=\\frac{2\\times10^{-11}\\text{ grams }}{10^{-6}\\text{ meters }^3}\\\\\\text{ }\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=2\\times10^{-11-\\left(-6\\right)}\\frac{\\text{ grams }}{\\text{ meters }^3}\\\\\\text{ }\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=2\\times10^{-5}\\frac{\\text{ grams }}{\\text{ meters }^3}\\\\\\end{array}[\/latex]<\/p>\n<p>If scientists know the density of healthy cells, they can compare the density of a sick person&#8217;s&nbsp;cells to that to rule out or test for disorders or diseases that may affect cellular density.<\/p>\n<h4 style=\"text-align: left;\">Answer<\/h4>\n<p style=\"text-align: left;\">The average density of a human cell is [latex]2\\times10^{-5}\\frac{\\text{ grams }}{\\text{ meters }^3}[\/latex]<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<p>The following video provides an example of how to find the number of operations a computer can perform in a very short amount of time.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Application of Scientific Notation - Quotient 2 (Time for Computer Operations)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Cbm6ejEbu-o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm88501\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=88501&theme=oea&iframe_resize_id=ohm88501&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"attachment_4381\" style=\"width: 352px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4381\" class=\"wp-image-4381\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/26223918\/Screen-Shot-2016-05-26-at-3.39.02-PM-300x137.png\" alt=\"Earth in the foreground, sun in the background, light beams traveling from the sun to the earth.\" width=\"342\" height=\"156\" \/><\/p>\n<p id=\"caption-attachment-4381\" class=\"wp-caption-text\">Light traveling from the sun to the earth.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the next example, you will use another well known formula, [latex]d=r\\cdot{t}[\/latex], to find how long it takes light to travel from the sun to the earth. Unlike the previous example, the distance between the earth and the sun is massive, so the numbers you will work with have positive exponents.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The speed of light is [latex]3\\times10^{8}\\frac{\\text{ meters }}{\\text{ second }}[\/latex]. If the sun is [latex]1.5\\times10^{11}[\/latex] meters from earth, how many seconds does it take for sunlight to reach the earth? &nbsp;Write your answer in scientific notation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q532092\">Show Solution<\/span><\/p>\n<div id=\"q532092\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Read and Understand:&nbsp;<\/strong>We are looking for how long\u2014an amount of time. We are given a rate which has units of meters per second and a distance in meters. This is a [latex]d=r\\cdot{t}[\/latex] problem.<\/p>\n<p><strong>Define and Translate:&nbsp;<\/strong><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}d=1.5\\times10^{11}\\\\r=3\\times10^{8}\\frac{\\text{ meters }}{\\text{ second }}\\\\t=\\text{ ? }\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\"><strong>Write and Solve:&nbsp;<\/strong>Substitute the values we are given into the&nbsp;[latex]d=r\\cdot{t}[\/latex] equation. We will work without units to make it easier. Often, scientists will work with units to make sure they have made correct calculations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}d=r\\cdot{t}\\\\1.5\\times10^{11}=3\\times10^{8}\\cdot{t}\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Divide both sides of the equation by&nbsp;[latex]3\\times10^{8}[\/latex] to isolate&nbsp;<em>t.<\/em><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}1.5\\times10^{11}=3\\times10^{8}\\cdot{t}\\\\\\text{ }\\\\\\frac{1.5\\times10^{11}}{3\\times10^{8}}=\\frac{3\\times10^{8}}{3\\times10^{8}}\\cdot{t}\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">On the left side, you will need to use the quotient rule of exponents to simplify, and on the right, you are left with&nbsp;<em>t.&nbsp;<\/em><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{1.5\\times10^{11}}{3\\times10^{8}}=\\frac{3\\times10^{8}}{3\\times10^{8}}\\cdot{t}\\\\\\text{ }\\\\\\left(\\frac{1.5}{3}\\right)\\times\\left(\\frac{10^{11}}{10^{8}}\\right)=t\\\\\\text{ }\\\\\\left(0.5\\right)\\times\\left(10^{11-8}\\right)=t\\\\0.5\\times10^3=t\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">This answer is not in scientific notation, so we will move the decimal to the right, which means we need to subtract one factor of [latex]10[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]0.5\\times10^3=5.0\\times10^2=t[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The time it takes light to travel from the sun to the earth is [latex]5.0\\times10^2=t[\/latex] seconds, or in standard&nbsp;notation, [latex]500[\/latex] seconds. &nbsp;That&#8217;s not bad considering how far it has to travel!<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the following video&nbsp;we calculate how many miles the participants of the New York marathon ran combined, and compare that to the circumference of the earth.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-2\" title=\"Application of Scientific Notation - Quotient 1 (Number of Times Around the Earth)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/15tw4-v100Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Summary<\/h3>\n<p>Scientific notation was developed to assist mathematicians, scientists, and others when expressing and working with very large and very small numbers. Scientific notation follows a very specific format in which a number is expressed as the product of a number greater than or equal to one and less than ten, and a power of [latex]10[\/latex]. The format is written [latex]a\\times10^{n}[\/latex], where [latex]1\\leq{a}<10[\/latex]&nbsp;and <i>n <\/i>is an integer.&nbsp;To multiply or divide numbers in scientific notation, you can use the commutative and associative properties to group the exponential terms together and apply the rules of exponents.<\/p>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a href=\"https:\/\/docs.google.com\/document\/d\/1sk-n1xsrRJlGvWjNjjTUYARfEfi42MnvFu8_NqY9HRI\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-15340\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Application of Scientific Notation - Quotient 1 (Number of Times Around the Earth). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/15tw4-v100Y\">https:\/\/youtu.be\/15tw4-v100Y<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Application of Scientific Notation - Quotient 2 (Time for Computer Operations). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Cbm6ejEbu-o\">https:\/\/youtu.be\/Cbm6ejEbu-o<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot: water molecule. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot: red blood cells. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot: light traveling from the sun to the earth. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-15340-1\">Orders of magnitude (mass). (n.d.). Retrieved May 26, 2016, from https:\/\/en.wikipedia.org\/wiki\/Orders_of_magnitude_(mass) <a href=\"#return-footnote-15340-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-15340-2\">How Big is a Human Cell? <a href=\"#return-footnote-15340-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-15340-3\">How big is a human cell? - Weizmann Institute of Science. (n.d.). Retrieved May 26, 2016, from http:\/\/www.weizmann.ac.il\/plants\/Milo\/images\/humanCellSize120116Clean.pdf <a href=\"#return-footnote-15340-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-15340-4\">Grover, W. H., Bryan, A. K., Diez-Silva, M., Suresh, S., Higgins, J. M., &amp; Manalis, S. R. (2011). Measuring single-cell density. Proceedings of the National Academy of Sciences, 108(27), 10992-10996. doi:10.1073\/pnas.1104651108 <a href=\"#return-footnote-15340-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":167848,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Application of Scientific Notation - Quotient 1 (Number of Times Around the Earth)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/15tw4-v100Y\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Application of Scientific Notation - Quotient 2 (Time for Computer Operations)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Cbm6ejEbu-o\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Screenshot: water molecule\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Screenshot: red blood cells\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Screenshot: light traveling from the sun to the earth\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"070e1afdb71d4fe699174d113cbbd5c3, 3a4db56a08504e8992112779baccdc3d","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15340","chapter","type-chapter","status-publish","hentry"],"part":15430,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15340","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/users\/167848"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15340\/revisions"}],"predecessor-version":[{"id":15528,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15340\/revisions\/15528"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/parts\/15430"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15340\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/media?parent=15340"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapter-type?post=15340"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/contributor?post=15340"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/license?post=15340"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}