{"id":295,"date":"2023-06-21T13:22:53","date_gmt":"2023-06-21T13:22:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/why-it-matters-exponential-and-logarithmic-equations-and-models-2\/"},"modified":"2023-07-09T04:05:59","modified_gmt":"2023-07-09T04:05:59","slug":"why-it-matters-exponential-and-logarithmic-equations-and-models-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/why-it-matters-exponential-and-logarithmic-equations-and-models-2\/","title":{"raw":"Putting It Together: Exponential and Logarithmic Equations and Models","rendered":"Putting It Together: Exponential and Logarithmic Equations and Models"},"content":{"raw":"\n\nAt the start of this module, you were assigned the task of analyzing a fossilized bone to determine its age. To make that estimate, you need to model for the decay rate of carbon-14.\n\nThe decay of a radioactive element is an exponential function of the form:\n

[latex]A\\left(t\\right)=A_0e^{-kt}[\/latex]<\/p>\n

where<\/p>\n

[latex]A(t)[\/latex] = mass of element remaining after t years<\/p>\n

[latex]A_0[\/latex] = original mass of element<\/p>\n

[latex]k[\/latex] = rate of decay<\/p>\n

[latex]t[\/latex] = time in years<\/p>\nSo to create a model for the decay function of carbon-14, assume for simplicity that the sample you started with had a mass of 1g. We know that half the starting mass of the sample will remain after one half-life which is 5,730 years. We can substitute these values for [latex]A(t)[\/latex] and [latex]A_0[\/latex] as follows:\n

[latex]A\\left(t\\right)=A_0e^{-kt}[\/latex]<\/p>\n

[latex]\\frac{1}{2}=(1)e^{-k\\left(5730\\right)}[\/latex]<\/p>\n

[latex]1n\\left(\\frac{1}{2}\\right)=1n\\left(e^{-k\\left(5730\\right)}\\right)[\/latex]<\/p>\n

[latex]1n\\left(2^{-1}\\right)=\\left(-5730k\\right)1n\\left(e\\right)[\/latex]<\/p>\n

[latex]-1n\\left(2\\right)=-5730k\\left(1\\right)[\/latex]<\/p>\n

[latex]k\\approx1.21\\times10^{-4}[\/latex]<\/p>\nNow you know the decay rate so you can write the equation for the exponential decay of carbon-14 and you can represent it as a graph.\n\n\"graph\n\nThe next step is to evaluate the function for a given mass. Assume a starting mass of 100 grams and that there are 20 grams remaining. Substitute these values into the model in the following way:\n

\n\n\n\n\n\n\n\n\n
Write the equation<\/td>\n[latex]A(t)=100e^{\\large{-(0.000121)t}}[\/latex]<\/td>\n<\/tr>\n
Substitute 20 grams for A(t)<\/td>\n[latex]20=100e^{\\large{-\\left(0.000121\\right)t}}[\/latex]<\/td>\n<\/tr>\n
Divide both sides by 100<\/td>\n[latex]0.20=e^{\\large{-\\left(0.000121\\right)t}}[\/latex]<\/td>\n<\/tr>\n
Change to logarithmic form<\/td>\n[latex]1n\\left(0.20\\right)=-\\left(0.000121\\right)t[\/latex]<\/td>\n<\/tr>\n
Divide both sides by -0.000121<\/td>\n[latex]t={\\large\\frac{1n\\left(0.20\\right)}{-0.000121}}[\/latex]<\/td>\n<\/tr>\n
Solve<\/td>\n[latex]t\\approx13,301[\/latex] years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\nNow you know that it would take 13,301 years for a 100-gram sample of carbon-14 to decay to the point that only 20 grams are left. Confirm that this number makes sense by looking at the graph.\n\nYou can also determine the amount of a 100-gram sample that would remain after a given number of years such as 8,000.  To do this, substitute the number of years into the function and evaluate.\n

[latex]A\\left(t\\right)=100e^{-\\left(0.000121\\right)t}[\/latex]<\/p>\n

[latex]A\\left(8000\\right)=100e^{-\\left(0.000121\\right)\\left(8000\\right)}\\approx38[\/latex] grams<\/p>\nAbout 38 grams would remain after 8,000 years.\n\nUnderstanding exponential functions helps scientists better understand radioactive decay and provides insights into past civilizations and species.\n\n","rendered":"

At the start of this module, you were assigned the task of analyzing a fossilized bone to determine its age. To make that estimate, you need to model for the decay rate of carbon-14.<\/p>\n

The decay of a radioactive element is an exponential function of the form:<\/p>\n

[latex]A\\left(t\\right)=A_0e^{-kt}[\/latex]<\/p>\n

where<\/p>\n

[latex]A(t)[\/latex] = mass of element remaining after t years<\/p>\n

[latex]A_0[\/latex] = original mass of element<\/p>\n

[latex]k[\/latex] = rate of decay<\/p>\n

[latex]t[\/latex] = time in years<\/p>\n

So to create a model for the decay function of carbon-14, assume for simplicity that the sample you started with had a mass of 1g. We know that half the starting mass of the sample will remain after one half-life which is 5,730 years. We can substitute these values for [latex]A(t)[\/latex] and [latex]A_0[\/latex] as follows:<\/p>\n

[latex]A\\left(t\\right)=A_0e^{-kt}[\/latex]<\/p>\n

[latex]\\frac{1}{2}=(1)e^{-k\\left(5730\\right)}[\/latex]<\/p>\n

[latex]1n\\left(\\frac{1}{2}\\right)=1n\\left(e^{-k\\left(5730\\right)}\\right)[\/latex]<\/p>\n

[latex]1n\\left(2^{-1}\\right)=\\left(-5730k\\right)1n\\left(e\\right)[\/latex]<\/p>\n

[latex]-1n\\left(2\\right)=-5730k\\left(1\\right)[\/latex]<\/p>\n

[latex]k\\approx1.21\\times10^{-4}[\/latex]<\/p>\n

Now you know the decay rate so you can write the equation for the exponential decay of carbon-14 and you can represent it as a graph.<\/p>\n

\"graph<\/p>\n

The next step is to evaluate the function for a given mass. Assume a starting mass of 100 grams and that there are 20 grams remaining. Substitute these values into the model in the following way:<\/p>\n

\n\n\n\n\n\n\n\n\n
Write the equation<\/td>\n[latex]A(t)=100e^{\\large{-(0.000121)t}}[\/latex]<\/td>\n<\/tr>\n
Substitute 20 grams for A(t)<\/td>\n[latex]20=100e^{\\large{-\\left(0.000121\\right)t}}[\/latex]<\/td>\n<\/tr>\n
Divide both sides by 100<\/td>\n[latex]0.20=e^{\\large{-\\left(0.000121\\right)t}}[\/latex]<\/td>\n<\/tr>\n
Change to logarithmic form<\/td>\n[latex]1n\\left(0.20\\right)=-\\left(0.000121\\right)t[\/latex]<\/td>\n<\/tr>\n
Divide both sides by -0.000121<\/td>\n[latex]t={\\large\\frac{1n\\left(0.20\\right)}{-0.000121}}[\/latex]<\/td>\n<\/tr>\n
Solve<\/td>\n[latex]t\\approx13,301[\/latex] years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

Now you know that it would take 13,301 years for a 100-gram sample of carbon-14 to decay to the point that only 20 grams are left. Confirm that this number makes sense by looking at the graph.<\/p>\n

You can also determine the amount of a 100-gram sample that would remain after a given number of years such as 8,000.  To do this, substitute the number of years into the function and evaluate.<\/p>\n

[latex]A\\left(t\\right)=100e^{-\\left(0.000121\\right)t}[\/latex]<\/p>\n

[latex]A\\left(8000\\right)=100e^{-\\left(0.000121\\right)\\left(8000\\right)}\\approx38[\/latex] grams<\/p>\n

About 38 grams would remain after 8,000 years.<\/p>\n

Understanding exponential functions helps scientists better understand radioactive decay and provides insights into past civilizations and species.<\/p>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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CC licensed content, Original<\/div>