{"id":184,"date":"2021-02-03T22:30:43","date_gmt":"2021-02-03T22:30:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=184"},"modified":"2021-05-27T18:56:48","modified_gmt":"2021-05-27T18:56:48","slug":"putting-it-together-functions-and-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/putting-it-together-functions-and-graphs\/","title":{"raw":"Putting It Together: Functions and Graphs","rendered":"Putting It Together: Functions and Graphs"},"content":{"raw":"

The Richter Scale for Earthquakes<\/h3>\r\n

In 1935, Charles Richter developed a scale (now known as the Richter scale<\/em><\/span>) to measure the magnitude of an earthquake<\/span>. The scale is a base-10 logarithmic scale, and it can be described as follows: Consider one earthquake with magnitude [latex]R_1[\/latex] on the Richter scale and a second earthquake with magnitude [latex]R_2[\/latex] on the Richter scale. Suppose [latex]R_1 > R_2[\/latex], which means the earthquake of magnitude [latex]R_1[\/latex] is stronger, but how much stronger is it than the other earthquake? A way of measuring the intensity of an earthquake is by using a seismograph to measure the amplitude of the earthquake waves. If [latex]A_1[\/latex] is the amplitude measured for the first earthquake and [latex]A_2[\/latex] is the amplitude measured for the second earthquake, then the amplitudes and magnitudes of the two earthquakes satisfy the following equation:<\/p>\r\n\r\n

[latex]R_1 - R_2 = \\log_{10}\\left(\\frac{A_1}{A_2}\\right)[\/latex]<\/div>\r\n

Consider an earthquake that measures 8 on the Richter scale and an earthquake that measures 7 on the Richter scale. Then,<\/p>\r\n\r\n

[latex]8-7=\\log_{10}\\left(\\frac{A_1}{A_2}\\right)[\/latex]<\/div>\r\n

Therefore,<\/p>\r\n\r\n

[latex]\\log_{10}\\left(\\frac{A_1}{A_2}\\right)=1[\/latex],<\/div>\r\n

which implies [latex]A_1 \/ A_2 = 10[\/latex] or [latex]A_1 = 10A_2[\/latex]. Since [latex]A_1[\/latex] is 10 times the size of [latex]A_2[\/latex], we say that the first earthquake is 10 times as intense as the second earthquake. On the other hand, if one earthquake measures 8 on the Richter scale and another measures 6, then the relative intensity of the two earthquakes satisfies the equation<\/p>\r\n\r\n

[latex]\\log_{10}\\left(\\frac{A_1}{A_2}\\right)=8-6=2[\/latex]<\/div>\r\n

Therefore, [latex]A_1=100A_2[\/latex]. That is, the first earthquake is 100 times more intense than the second earthquake.<\/p>\r\n

How can we use logarithmic functions to compare the relative severity of the magnitude 9 earthquake in Japan in 2011 with the magnitude 7.3 earthquake in Haiti in 2010?<\/p>\r\n

To compare the Japan and Haiti earthquakes, we can use an equation presented earlier:<\/p>\r\n

[latex]9-7.3=\\log_{10}\\left(\\frac{A_1}{A_2}\\right)[\/latex]<\/p>\r\n

Therefore, [latex]\\frac{A_1}{A_2}=10^{1.7}[\/latex], and we conclude that the earthquake in Japan was approximately 50 times more intense than the earthquake in Haiti.<\/p>","rendered":"

The Richter Scale for Earthquakes<\/h3>\n

In 1935, Charles Richter developed a scale (now known as the Richter scale<\/em><\/span>) to measure the magnitude of an earthquake<\/span>. The scale is a base-10 logarithmic scale, and it can be described as follows: Consider one earthquake with magnitude [latex]R_1[\/latex] on the Richter scale and a second earthquake with magnitude [latex]R_2[\/latex] on the Richter scale. Suppose [latex]R_1 > R_2[\/latex], which means the earthquake of magnitude [latex]R_1[\/latex] is stronger, but how much stronger is it than the other earthquake? A way of measuring the intensity of an earthquake is by using a seismograph to measure the amplitude of the earthquake waves. If [latex]A_1[\/latex] is the amplitude measured for the first earthquake and [latex]A_2[\/latex] is the amplitude measured for the second earthquake, then the amplitudes and magnitudes of the two earthquakes satisfy the following equation:<\/p>\n

[latex]R_1 - R_2 = \\log_{10}\\left(\\frac{A_1}{A_2}\\right)[\/latex]<\/div>\n

Consider an earthquake that measures 8 on the Richter scale and an earthquake that measures 7 on the Richter scale. Then,<\/p>\n

[latex]8-7=\\log_{10}\\left(\\frac{A_1}{A_2}\\right)[\/latex]<\/div>\n

Therefore,<\/p>\n

[latex]\\log_{10}\\left(\\frac{A_1}{A_2}\\right)=1[\/latex],<\/div>\n

which implies [latex]A_1 \/ A_2 = 10[\/latex] or [latex]A_1 = 10A_2[\/latex]. Since [latex]A_1[\/latex] is 10 times the size of [latex]A_2[\/latex], we say that the first earthquake is 10 times as intense as the second earthquake. On the other hand, if one earthquake measures 8 on the Richter scale and another measures 6, then the relative intensity of the two earthquakes satisfies the equation<\/p>\n

[latex]\\log_{10}\\left(\\frac{A_1}{A_2}\\right)=8-6=2[\/latex]<\/div>\n

Therefore, [latex]A_1=100A_2[\/latex]. That is, the first earthquake is 100 times more intense than the second earthquake.<\/p>\n

How can we use logarithmic functions to compare the relative severity of the magnitude 9 earthquake in Japan in 2011 with the magnitude 7.3 earthquake in Haiti in 2010?<\/p>\n

To compare the Japan and Haiti earthquakes, we can use an equation presented earlier:<\/p>\n

[latex]9-7.3=\\log_{10}\\left(\\frac{A_1}{A_2}\\right)[\/latex]<\/p>\n

Therefore, [latex]\\frac{A_1}{A_2}=10^{1.7}[\/latex], and we conclude that the earthquake in Japan was approximately 50 times more intense than the earthquake in Haiti.<\/p>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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