{"id":155,"date":"2026-04-10T02:53:13","date_gmt":"2026-04-10T02:53:13","guid":{"rendered":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/chapter\/ch071-xhtml\/"},"modified":"2026-05-12T01:57:46","modified_gmt":"2026-05-12T01:57:46","slug":"solution-exercise-4","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/chapter\/solution-exercise-4\/","title":{"raw":"Solution Exercise\u00a04","rendered":"Solution Exercise\u00a04"},"content":{"raw":"<div id=\"solution-exercise-4\" class=\"level1\">\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe solution to Exercise\u00a04 has two parts: a) and b).\r\n\r\n<\/div>\r\n<ol type=\"a\">\r\n \t<li>Using Equation\u00a0<a href=\"\/age-dating-young-groundwater\/chapter\/3-4-calculation-of-tracer-age\/#equation-13\"><span data-custom-style=\"Hyperlink\">(13)<\/span><\/a> the minimum groundwater age is determined as:<\/li>\r\n<\/ol>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe average precipitation\u2011weighted [latex]_{}^{3}H[\/latex] for the last five\u00a0years is 7.7\u00a0TU.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote><em>Age<sub>min<\/sub>\u00a0<\/em>=[latex]\\ -[\/latex]ln(4.5\/7.7)\/(ln(2)\/12.32))<em>\u00a0<\/em>= 9 years (rounded down to capture a minimum)<\/blockquote>\r\n<\/div>\r\n<ol start=\"2\" type=\"a\">\r\n \t<li>Based on the plotted decay curve and [latex]_{}^{3}H[\/latex] function, we can estimate a minimum groundwater age of nine\u00a0years and a range of potential minimum age between four and 20\u00a0years.<\/li>\r\n<\/ol>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe graphical solution steps, based on information in Section\u00a03.4, are:\r\n\r\n<\/div>\r\n<ol type=\"1\">\r\n \t<li>Use the provided atmospheric [latex]_{}^{3}H[\/latex] function.\r\n<ul>\r\n \t<li>The data provided are annual averages and the problem statement indicates we can assume these are weighted by precipitation.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Calculate the first point of the decay curve:<\/li>\r\n<\/ol>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>Year<em>\u00a0<\/em>= 2012<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote><em>C<sub>m<\/sub>\u00a0<\/em>= 4.5 TU<\/blockquote>\r\n<\/div>\r\n<ol start=\"3\" type=\"1\">\r\n \t<li>Calculate the second point of the decay curve:<\/li>\r\n<\/ol>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote><em>C<sub>0<\/sub>\u00a0<\/em>= <em>C<sub>m<\/sub><\/em>(exp(\u03bb\u0394t)<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote><em>C<sub>0<\/sub>\u00a0<\/em>= 4.5 TU (exp[latex]\\left( \\frac{ln(2)}{12.32\\ yrs}(2012\\ yrs - 1950\\ yrs) \\right)[\/latex]<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>Year<em>\u00a0<\/em>= 1950<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote><em>C<sub>0<\/sub>\u00a0<\/em>= 147 TU<\/blockquote>\r\n<\/div>\r\n<ol start=\"4\" type=\"1\">\r\n \t<li>Therefore, the two points for plotting the decay curve are:<\/li>\r\n<\/ol>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>Year : [latex]_{}^{3}H[\/latex] (TU)<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>2012 : 4.5 TU<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>1950 : 147 TU<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nMore detail is provided in the following graphs and the <a href=\"https:\/\/gw-project.org\/books\/age-dating-young-groundwater\/\"><span data-custom-style=\"Hyperlink\">spreadsheet provided on the web page for this book<\/span><\/a>\u2197. Based on the plotted decay curve and the precipitation\u2011weighted [latex]_{}^{3}H[\/latex] curve, the estimated minimum groundwater age is 2012<em>\u2212<\/em>2003<em>\u00a0<\/em>= 9 years. Using the mean annual [latex]_{}^{3}H[\/latex] curve, plus or minus one standard deviation, we can estimate uncertainty in minimum groundwater age. The decay curve crosses the annual mean minus one standard deviation line at about 2008. The decay curve crosses the annual mean plus one standard deviation line at about 1992. Thus, the full range of potential minimum ages is about 4 to 20 years.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.83333in; height: 3.46111in;\" src=\"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-content\/uploads\/sites\/41\/2026\/04\/file33.png\" alt=\"\" \/>\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nA shorter period of the same graph, zoomed in on the period of time when there are intersections of the decay line and the recharge concentration curve.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.83333in; height: 3.64583in;\" src=\"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-content\/uploads\/sites\/41\/2026\/04\/file34.png\" alt=\"\" \/>\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<a href=\"\/age-dating-young-groundwater\/chapter\/exercise-4\/\"><span data-custom-style=\"Hyperlink\"><u>Return to Exercise\u00a04<\/u><\/span><\/a>\u2ba5\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<a href=\"\/age-dating-young-groundwater\/chapter\/3-5-sensitivity-of-age-to-input-parameters\/\"><span data-custom-style=\"Hyperlink\"><u>Return to where text linked to Exercise\u00a04<\/u><\/span><\/a>\u2ba5\r\n\r\n<\/div>\r\n<\/div>","rendered":"<div id=\"solution-exercise-4\" class=\"level1\">\n<div data-custom-style=\"First Paragraph\">\n<p>The solution to Exercise\u00a04 has two parts: a) and b).<\/p>\n<\/div>\n<ol type=\"a\">\n<li>Using Equation\u00a0<a href=\"\/age-dating-young-groundwater\/chapter\/3-4-calculation-of-tracer-age\/#equation-13\"><span data-custom-style=\"Hyperlink\">(13)<\/span><\/a> the minimum groundwater age is determined as:<\/li>\n<\/ol>\n<div data-custom-style=\"First Paragraph\">\n<p>The average precipitation\u2011weighted [latex]_{}^{3}H[\/latex] for the last five\u00a0years is 7.7\u00a0TU.<\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p><em>Age<sub>min<\/sub>\u00a0<\/em>=[latex]\\ -[\/latex]ln(4.5\/7.7)\/(ln(2)\/12.32))<em>\u00a0<\/em>= 9 years (rounded down to capture a minimum)<\/p><\/blockquote>\n<\/div>\n<ol start=\"2\" type=\"a\">\n<li>Based on the plotted decay curve and [latex]_{}^{3}H[\/latex] function, we can estimate a minimum groundwater age of nine\u00a0years and a range of potential minimum age between four and 20\u00a0years.<\/li>\n<\/ol>\n<div data-custom-style=\"First Paragraph\">\n<p>The graphical solution steps, based on information in Section\u00a03.4, are:<\/p>\n<\/div>\n<ol type=\"1\">\n<li>Use the provided atmospheric [latex]_{}^{3}H[\/latex] function.\n<ul>\n<li>The data provided are annual averages and the problem statement indicates we can assume these are weighted by precipitation.<\/li>\n<\/ul>\n<\/li>\n<li>Calculate the first point of the decay curve:<\/li>\n<\/ol>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>Year<em>\u00a0<\/em>= 2012<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p><em>C<sub>m<\/sub>\u00a0<\/em>= 4.5 TU<\/p><\/blockquote>\n<\/div>\n<ol start=\"3\" type=\"1\">\n<li>Calculate the second point of the decay curve:<\/li>\n<\/ol>\n<div data-custom-style=\"Block Text\">\n<blockquote><p><em>C<sub>0<\/sub>\u00a0<\/em>= <em>C<sub>m<\/sub><\/em>(exp(\u03bb\u0394t)<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p><em>C<sub>0<\/sub>\u00a0<\/em>= 4.5 TU (exp[latex]\\left( \\frac{ln(2)}{12.32\\ yrs}(2012\\ yrs - 1950\\ yrs) \\right)[\/latex]<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>Year<em>\u00a0<\/em>= 1950<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p><em>C<sub>0<\/sub>\u00a0<\/em>= 147 TU<\/p><\/blockquote>\n<\/div>\n<ol start=\"4\" type=\"1\">\n<li>Therefore, the two points for plotting the decay curve are:<\/li>\n<\/ol>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>Year : [latex]_{}^{3}H[\/latex] (TU)<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>2012 : 4.5 TU<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>1950 : 147 TU<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>More detail is provided in the following graphs and the <a href=\"https:\/\/gw-project.org\/books\/age-dating-young-groundwater\/\"><span data-custom-style=\"Hyperlink\">spreadsheet provided on the web page for this book<\/span><\/a>\u2197. Based on the plotted decay curve and the precipitation\u2011weighted [latex]_{}^{3}H[\/latex] curve, the estimated minimum groundwater age is 2012<em>\u2212<\/em>2003<em>\u00a0<\/em>= 9 years. Using the mean annual [latex]_{}^{3}H[\/latex] curve, plus or minus one standard deviation, we can estimate uncertainty in minimum groundwater age. The decay curve crosses the annual mean minus one standard deviation line at about 2008. The decay curve crosses the annual mean plus one standard deviation line at about 1992. Thus, the full range of potential minimum ages is about 4 to 20 years.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.83333in; height: 3.46111in;\" src=\"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-content\/uploads\/sites\/41\/2026\/04\/file33.png\" alt=\"\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>A shorter period of the same graph, zoomed in on the period of time when there are intersections of the decay line and the recharge concentration curve.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.83333in; height: 3.64583in;\" src=\"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-content\/uploads\/sites\/41\/2026\/04\/file34.png\" alt=\"\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><a href=\"\/age-dating-young-groundwater\/chapter\/exercise-4\/\"><span data-custom-style=\"Hyperlink\"><u>Return to Exercise\u00a04<\/u><\/span><\/a>\u2ba5<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><a href=\"\/age-dating-young-groundwater\/chapter\/3-5-sensitivity-of-age-to-input-parameters\/\"><span data-custom-style=\"Hyperlink\"><u>Return to where text linked to Exercise\u00a04<\/u><\/span><\/a>\u2ba5<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":7,"menu_order":49,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-155","chapter","type-chapter","status-publish","hentry"],"part":45,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/155","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/users\/7"}],"version-history":[{"count":4,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/155\/revisions"}],"predecessor-version":[{"id":439,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/155\/revisions\/439"}],"part":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/parts\/45"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/155\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/media?parent=155"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapter-type?post=155"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/contributor?post=155"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/license?post=155"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}