{"id":932,"date":"2016-12-21T21:00:46","date_gmt":"2016-12-21T21:00:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&p=932"},"modified":"2017-04-04T23:20:41","modified_gmt":"2017-04-04T23:20:41","slug":"putting-it-together-growth-models","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/putting-it-together-growth-models\/","title":{"raw":"Putting It Together: Growth Models","rendered":"Putting It Together: Growth Models"},"content":{"raw":"At the beginning of the module, you were introduced to some social media statistics for several celebrities. \u00a0Since then you learned that you could describe and compare growth by understanding a little bit about different types of growth, and using a growth model to make a prediction.\r\n\r\nNow how about you be the one tracking the celebrities? \u00a0Imagine that you started a celebrity blog a few years back. \u00a0You write about hot news in Hollywood, and what your favorite celebs are up to. \u00a0Between 2010 and 2012, your blog followers grew at a rate of 5% to 9,500 people. \u00a0You hope to have 20,000 followers by 2020. \u00a0If this growth rate continues, will you meet your goal?\r\n\r\n\"Photo<\/a>\r\n\r\n \r\n\r\nBecause you know that the number of followers is growing at 5%, you know you\u2019re dealing with exponential growth. \u00a0In order to develop a model, you need an initial value. To determine the initial value, you can use the number of followers in 2012 ([latex]P_0[\/latex] = 9,500), and the rate of growth, which is 5% ([latex]r[\/latex] = 5%), and solve for [latex]P_0[\/latex].\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]P_n=\\left(1+r\\right)^nP_0[\/latex]<\/td>\r\nStart with the explicit form of the exponential growth equation.<\/td>\r\n<\/tr>\r\n
[latex]P_n=\\left(1+0.05\\right)^nP_0[\/latex]<\/td>\r\nSubstitute the growth rate, 5%, written as a decimal for [latex]r[\/latex].<\/td>\r\n<\/tr>\r\n
[latex]P_n=\\left(1+0.05\\right)^n9,500[\/latex]<\/td>\r\nSubstitute the initial value, which is the number of followers in 2012, for [latex]P_0[\/latex].<\/td>\r\n<\/tr>\r\n
[latex]P_8=\\left(1+0.05\\right)^89,500[\/latex]<\/td>\r\nSubstitute the number of years between 2012 and 2020 for [latex]n[\/latex].<\/td>\r\n<\/tr>\r\n
[latex]P_8=\\left(1.05\\right)^89,500\\approx14,000[\/latex]<\/td>\r\nEvaluate.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n \r\n\r\nUnfortunately, you won\u2019t quite meet your goal, but you\u2019ll be close.\r\n\r\nWhile online, you happen to notice that a different celebrity blogger is experiencing amazing success. \u00a0She appeared on the scene only nine months ago, but in that short time it grew exponentially from an initial following of 5,000 people to 12,000. \u00a0At what rate of growth is her following increasing?\r\n\r\n \r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]P_0=5,000[\/latex]<\/td>\r\nDetermine the initial value.<\/td>\r\n<\/tr>\r\n
[latex]P_9=\\left(1+r\\right)^9P_0[\/latex]<\/td>\r\nAgain use the explicit form of the exponential growth equation, but this time [latex]t=9[\/latex] months.<\/td>\r\n<\/tr>\r\n
[latex]12,000=\\left(1+r\\right)^95,000[\/latex]<\/td>\r\nSubstitute the initial value, which is the number of followers when she started, for [latex]P_0[\/latex].<\/td>\r\n<\/tr>\r\n
[latex]{\\large\\frac{12,000}{5,000}}=\\left(1+r\\right)^9[\/latex]<\/td>\r\nDivide both sides by 5,000.<\/td>\r\n<\/tr>\r\n
[latex]\\sqrt[9]{12,000\/5,000}=(1+r)^9[\/latex]<\/td>\r\nTake the ninth root of both sides.<\/td>\r\n<\/tr>\r\n
[latex]r=\\sqrt[9]{12,000\/5,000}-1\\approx0.10[\/latex]<\/td>\r\nSolve for [latex]r[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n

<\/h2>\r\nSo her blog is growing at a rate of 10%. \u00a0What\u2019s she got that you don\u2019t have? \u00a0By comparing the rates of growth, you know it is definitely something. \u00a0Now you just have to figure out what it is and jazz up your blog to get followers breaking down the internet to read what you have to say!\r\n\r\n ","rendered":"

At the beginning of the module, you were introduced to some social media statistics for several celebrities. \u00a0Since then you learned that you could describe and compare growth by understanding a little bit about different types of growth, and using a growth model to make a prediction.<\/p>\n

Now how about you be the one tracking the celebrities? \u00a0Imagine that you started a celebrity blog a few years back. \u00a0You write about hot news in Hollywood, and what your favorite celebs are up to. \u00a0Between 2010 and 2012, your blog followers grew at a rate of 5% to 9,500 people. \u00a0You hope to have 20,000 followers by 2020. \u00a0If this growth rate continues, will you meet your goal?<\/p>\n

\"Photo<\/a><\/p>\n

 <\/p>\n

Because you know that the number of followers is growing at 5%, you know you\u2019re dealing with exponential growth. \u00a0In order to develop a model, you need an initial value. To determine the initial value, you can use the number of followers in 2012 ([latex]P_0[\/latex] = 9,500), and the rate of growth, which is 5% ([latex]r[\/latex] = 5%), and solve for [latex]P_0[\/latex].<\/p>\n

\n\n\n\n\n\n\n\n
[latex]P_n=\\left(1+r\\right)^nP_0[\/latex]<\/td>\nStart with the explicit form of the exponential growth equation.<\/td>\n<\/tr>\n
[latex]P_n=\\left(1+0.05\\right)^nP_0[\/latex]<\/td>\nSubstitute the growth rate, 5%, written as a decimal for [latex]r[\/latex].<\/td>\n<\/tr>\n
[latex]P_n=\\left(1+0.05\\right)^n9,500[\/latex]<\/td>\nSubstitute the initial value, which is the number of followers in 2012, for [latex]P_0[\/latex].<\/td>\n<\/tr>\n
[latex]P_8=\\left(1+0.05\\right)^89,500[\/latex]<\/td>\nSubstitute the number of years between 2012 and 2020 for [latex]n[\/latex].<\/td>\n<\/tr>\n
[latex]P_8=\\left(1.05\\right)^89,500\\approx14,000[\/latex]<\/td>\nEvaluate.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

 <\/p>\n

Unfortunately, you won\u2019t quite meet your goal, but you\u2019ll be close.<\/p>\n

While online, you happen to notice that a different celebrity blogger is experiencing amazing success. \u00a0She appeared on the scene only nine months ago, but in that short time it grew exponentially from an initial following of 5,000 people to 12,000. \u00a0At what rate of growth is her following increasing?<\/p>\n

 <\/p>\n

\n\n\n\n\n\n\n\n\n
[latex]P_0=5,000[\/latex]<\/td>\nDetermine the initial value.<\/td>\n<\/tr>\n
[latex]P_9=\\left(1+r\\right)^9P_0[\/latex]<\/td>\nAgain use the explicit form of the exponential growth equation, but this time [latex]t=9[\/latex] months.<\/td>\n<\/tr>\n
[latex]12,000=\\left(1+r\\right)^95,000[\/latex]<\/td>\nSubstitute the initial value, which is the number of followers when she started, for [latex]P_0[\/latex].<\/td>\n<\/tr>\n
[latex]{\\large\\frac{12,000}{5,000}}=\\left(1+r\\right)^9[\/latex]<\/td>\nDivide both sides by 5,000.<\/td>\n<\/tr>\n
[latex]\\sqrt[9]{12,000\/5,000}=(1+r)^9[\/latex]<\/td>\nTake the ninth root of both sides.<\/td>\n<\/tr>\n
[latex]r=\\sqrt[9]{12,000\/5,000}-1\\approx0.10[\/latex]<\/td>\nSolve for [latex]r[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

<\/h2>\n

So her blog is growing at a rate of 10%. \u00a0What\u2019s she got that you don\u2019t have? \u00a0By comparing the rates of growth, you know it is definitely something. \u00a0Now you just have to figure out what it is and jazz up your blog to get followers breaking down the internet to read what you have to say!<\/p>\n

 <\/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>