{"id":1789,"date":"2024-05-27T14:20:43","date_gmt":"2024-05-27T18:20:43","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=1789"},"modified":"2024-06-12T14:49:25","modified_gmt":"2024-06-12T18:49:25","slug":"solving-for-x","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/solving-for-x\/","title":{"raw":"Solving for X","rendered":"Solving for X"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nCalculate the missing [latex]x[\/latex] when given a probability in continuous uniform problems.\r\n\r\n<\/div>\r\n<\/div>\r\nWe can solve for [latex]x[\/latex] when given a probability, [latex]p[\/latex]:\r\n<ul>\r\n \t<li>Use [latex]p = P(x_1 \\le x \\le x_2) =\u00a0 \\frac{x_2 - x_1}{b-a}[\/latex]<\/li>\r\n \t<li>And, use algebra to solve for the missing [latex]x[\/latex] value.<\/li>\r\n<\/ul>\r\n<h1>Travel Times (at least ExERCISE)<\/h1>\r\nLet us revisit the commute times example and calculate the [latex]x[\/latex]-value when given the probability of at least [latex]x[\/latex].\r\n<h2>Example 36.1.1<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: During the morning commute, the time it takes to drive to BCIT follows a uniform distribution and is between 30 and 55 minutes.\r\n\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: 50% of the time it takes you at least how long to get to campus?\r\n\r\n<span style=\"color: #003366\"><strong>You Try<\/strong><\/span>: Let us first determine the values to input in the equation below:\r\n\r\n\\[ P(x_1 \\le x \\le x_2) = \\frac{x_2-x_1}{b-a} = p \\]\r\n\r\n[h5p id=\"98\"]\r\n\r\n<span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: Now, use algebra to solve for the minimum time it takes you half of the time to get to campus. Select your answer from the options below:\r\n\r\n[h5p id=\"99\"]\r\n<div>\r\n<h1>Click below to reveal the full solutions for the above exercise.<\/h1>\r\nFirst, the values we input are the following:\r\n<ul>\r\n \t<li>[latex]a = 30[\/latex] (minimum possible travel time)<\/li>\r\n \t<li>[latex]b = 55[\/latex] (maximum possible travel time)<\/li>\r\n \t<li>[latex]x_1 = X[\/latex] (we do not know this minimum [latex]x[\/latex]-value)<\/li>\r\n \t<li>[latex]x_2 = 55[\/latex] (we know the probability of at least a certain travel time - so there is no upper limit)<\/li>\r\n \t<li>[latex]p = 50\\% = 0.5[\/latex] (50% of the time it takes you at least how long?)<\/li>\r\n<\/ul>\r\nLet us plug them into the formula [latex]P(x_1 \\le x \\le x_2) = \\frac{x_2-x_1}{b-a} = p [\/latex] and solve for [latex]X[\/latex]:\r\n\r\n[latex]\r\n\r\n\\begin{align}\r\n\r\nP(X \\le x \\le 55) = \\frac{55-X}{55-30} &amp;= 0.5 \\\\\r\n\r\n55-X &amp;= 0.5 \\times (55-30) \\\\\r\n\r\n55-X &amp;= 0.5 \\times 25 = 12.5 \\\\\r\n\r\n-X &amp;= 12.5-55 \\\\\r\n\r\n-X &amp;= -42.5 \\\\\r\n\r\nX &amp;= 42.5 \\\\\r\n\r\n\\end{align}\r\n\r\n[\/latex]\r\n\r\n<\/div>\r\n<h1>Travel Times Written Example (Less Than Video)<\/h1>\r\nLet us now change up the problem slightly and show the solution in a video (see below).\r\n<h2>Example 36.1.2<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Let us continue on with the previous example:\r\n<ul>\r\n \t<li>Your commute times follow a uniform distribution<\/li>\r\n \t<li>And are between 30 and 55 minutes.<\/li>\r\n<\/ul>\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: 30% of the time, it takes you less than how many minutes to get to campus? Solve the problem by hand.\r\n\r\n<strong><span style=\"color: #003366\">Solution<\/span><\/strong>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/Continuous-Uniform-Solving-for-X-Value.pdf\">Click here<\/a>\u00a0to download the written solutions. Also, see the video below:\r\n\r\nhttps:\/\/youtu.be\/-cMklCIg7BM\r\n<h1>Travel Times Using Excel's Goal Seek (Video)<\/h1>\r\nLet us finish up this section by going through how to use Excel's Goal Seek to solve the previous example (see below).\r\n<h2>Example 36.1.3<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Let us continue on with the previous example:\r\n<ul>\r\n \t<li>Your commute times follow a uniform distribution<\/li>\r\n \t<li>And are between 30 and 55 minutes.<\/li>\r\n<\/ul>\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: 30% of the time, it takes you less than how many minutes to get to campus? Solve the problem using <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/use-goal-seek-to-find-the-result-you-want-by-adjusting-an-input-value-320cb99e-f4a4-417f-b1c3-4f369d6e66c7\">Excel's Goal Seek<\/a>.\r\n\r\n<strong><span style=\"color: #003366\">Solution<\/span><\/strong>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/Example_36-1-3.xlsx\">Click here<\/a> to download the Excel solutions shown in the solution video below:\r\n\r\nhttps:\/\/youtu.be\/GxIeBu-fI5o\r\n<h1>Key Takeaways (EXERCISE)<\/h1>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways: Solving for X<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[h5p id=\"100\"]\r\n\r\n[h5p id=\"101\"]\r\n\r\n<\/div>\r\n<\/div>\r\n<h1>Your Own Notes (EXERCISE)<\/h1>\r\n<ul>\r\n \t<li>Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t<li>These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t<li>Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"16\"]","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Calculate the missing [latex]x[\/latex] when given a probability in continuous uniform problems.<\/p>\n<\/div>\n<\/div>\n<p>We can solve for [latex]x[\/latex] when given a probability, [latex]p[\/latex]:<\/p>\n<ul>\n<li>Use [latex]p = P(x_1 \\le x \\le x_2) =\u00a0 \\frac{x_2 - x_1}{b-a}[\/latex]<\/li>\n<li>And, use algebra to solve for the missing [latex]x[\/latex] value.<\/li>\n<\/ul>\n<h1>Travel Times (at least ExERCISE)<\/h1>\n<p>Let us revisit the commute times example and calculate the [latex]x[\/latex]-value when given the probability of at least [latex]x[\/latex].<\/p>\n<h2>Example 36.1.1<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: During the morning commute, the time it takes to drive to BCIT follows a uniform distribution and is between 30 and 55 minutes.<\/p>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: 50% of the time it takes you at least how long to get to campus?<\/p>\n<p><span style=\"color: #003366\"><strong>You Try<\/strong><\/span>: Let us first determine the values to input in the equation below:<\/p>\n<p>\\[ P(x_1 \\le x \\le x_2) = \\frac{x_2-x_1}{b-a} = p \\]<\/p>\n<div id=\"h5p-98\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-98\" class=\"h5p-iframe\" data-content-id=\"98\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 35.1.1 - Uniform - values of a, b, x1 and x2\"><\/iframe><\/div>\n<\/div>\n<p><span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: Now, use algebra to solve for the minimum time it takes you half of the time to get to campus. Select your answer from the options below:<\/p>\n<div id=\"h5p-99\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-99\" class=\"h5p-iframe\" data-content-id=\"99\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 36.1.1 - Solving for X, At Least Example, Uniform Distributions\"><\/iframe><\/div>\n<\/div>\n<div>\n<h1>Click below to reveal the full solutions for the above exercise.<\/h1>\n<p>First, the values we input are the following:<\/p>\n<ul>\n<li>[latex]a = 30[\/latex] (minimum possible travel time)<\/li>\n<li>[latex]b = 55[\/latex] (maximum possible travel time)<\/li>\n<li>[latex]x_1 = X[\/latex] (we do not know this minimum [latex]x[\/latex]-value)<\/li>\n<li>[latex]x_2 = 55[\/latex] (we know the probability of at least a certain travel time &#8211; so there is no upper limit)<\/li>\n<li>[latex]p = 50\\% = 0.5[\/latex] (50% of the time it takes you at least how long?)<\/li>\n<\/ul>\n<p>Let us plug them into the formula [latex]P(x_1 \\le x \\le x_2) = \\frac{x_2-x_1}{b-a} = p[\/latex] and solve for [latex]X[\/latex]:<\/p>\n<p>[latex]\\begin{align}    P(X \\le x \\le 55) = \\frac{55-X}{55-30} &= 0.5 \\\\    55-X &= 0.5 \\times (55-30) \\\\    55-X &= 0.5 \\times 25 = 12.5 \\\\    -X &= 12.5-55 \\\\    -X &= -42.5 \\\\    X &= 42.5 \\\\    \\end{align}[\/latex]<\/p>\n<\/div>\n<h1>Travel Times Written Example (Less Than Video)<\/h1>\n<p>Let us now change up the problem slightly and show the solution in a video (see below).<\/p>\n<h2>Example 36.1.2<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Let us continue on with the previous example:<\/p>\n<ul>\n<li>Your commute times follow a uniform distribution<\/li>\n<li>And are between 30 and 55 minutes.<\/li>\n<\/ul>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: 30% of the time, it takes you less than how many minutes to get to campus? Solve the problem by hand.<\/p>\n<p><strong><span style=\"color: #003366\">Solution<\/span><\/strong>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/Continuous-Uniform-Solving-for-X-Value.pdf\">Click here<\/a>\u00a0to download the written solutions. Also, see the video below:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Solving for an unknown in Continuous Uniform Distributions\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/-cMklCIg7BM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h1>Travel Times Using Excel&#8217;s Goal Seek (Video)<\/h1>\n<p>Let us finish up this section by going through how to use Excel&#8217;s Goal Seek to solve the previous example (see below).<\/p>\n<h2>Example 36.1.3<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Let us continue on with the previous example:<\/p>\n<ul>\n<li>Your commute times follow a uniform distribution<\/li>\n<li>And are between 30 and 55 minutes.<\/li>\n<\/ul>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: 30% of the time, it takes you less than how many minutes to get to campus? Solve the problem using <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/use-goal-seek-to-find-the-result-you-want-by-adjusting-an-input-value-320cb99e-f4a4-417f-b1c3-4f369d6e66c7\">Excel&#8217;s Goal Seek<\/a>.<\/p>\n<p><strong><span style=\"color: #003366\">Solution<\/span><\/strong>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/Example_36-1-3.xlsx\">Click here<\/a> to download the Excel solutions shown in the solution video below:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"How to solve for an unknown value using Goal Seek for a continuous uniform example.\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GxIeBu-fI5o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h1>Key Takeaways (EXERCISE)<\/h1>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways: Solving for X<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"h5p-100\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-100\" class=\"h5p-iframe\" data-content-id=\"100\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Continuous Uniform - Solving for X - Key Takeaways\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-101\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-101\" class=\"h5p-iframe\" data-content-id=\"101\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Continuous Uniform - Solving for an unknown value - Key Takeaways Solutions\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Your Own Notes (EXERCISE)<\/h1>\n<ul>\n<li>Are there any notes you want to take from this section? Is there anything you&#8217;d like to copy and paste below?<\/li>\n<li>These notes are for you only (they will not be stored anywhere)<\/li>\n<li>Make sure to download them at the end to use as a reference<\/li>\n<\/ul>\n<div id=\"h5p-16\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-16\" class=\"h5p-iframe\" data-content-id=\"16\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key takeaways, notes and comments from this section document tool.\"><\/iframe><\/div>\n<\/div>\n","protected":false},"author":865,"menu_order":8,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1789","chapter","type-chapter","status-publish","hentry"],"part":1701,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1789","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/users\/865"}],"version-history":[{"count":25,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1789\/revisions"}],"predecessor-version":[{"id":1963,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1789\/revisions\/1963"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/1701"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1789\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=1789"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1789"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=1789"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=1789"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}