Questions and Answers #1 Exponential Models

Questions and Answers Sheet 1 Exponential Models Question #1 You open a bank account to save for college and deposit $400 in the account. Each year, the balance in your account will increase 5%. a. Write a function that models your annual balance. b. What will be the total amount in your account after 7 yr? Use the exponential function and extend the table to answer part b. Answer: Years account Open Balance this Year Total Balance 0 $400 $400 1 $400(1.05) $420 2 $420(1.05) $441 3 $441(1.05) $463.05 4 $463.05(1.05) $486.20 5 $486.20(1.05) $510.51 6 $510.51(1.05) $536.04 7 $536.04(1.05) $562.84 So, the total balance after 7 years is $562.84 Question #2 For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places. x 0 0.5 1 1.5 2 3 4 5 6 7 8 f ( x) 2.2 2.9 3.9 4.8 6.4 9.3 12.3 15 16.2 17.3 17.9 Answer: Step 1 Remember that regression analysis is the process of looking for a best fit of model for a set of data. This can be done on a graphing utility as follows: 1. Press [STAT], the input corresponging x-values of data in L1, and y-values of data in L2. 2. Use [STATPLOT] to observe a scatterplot of the data. 3. Press [STAT], then [CALC] then [ExpReg]/[LnReg]/[Logistic]. This will show you a function in either the form of an exponential, a logarithmic or a logistic model. 4. Graph this equation on the same window as the scatterplot to see if it fits the data. Step 2 1. Press [STAT], the input corresponging x-values of data in L1, and y-values of data in L2. 2. Use [STATPLOT] to observe a scatterplot of the data. Step 3 Based on the plots of the points, it can be exponential or logarithmic. However, upon checking both regression analysis, the one with the closest value of 𝑟 2 to 1 is 18.41663 exponential, hence, its formula is 𝑦 = −0.68374𝑥 The graph of which is below: 1+7.54619𝑒 Answer: 𝑦 = 18.41663 1+7.54619𝑒 −0.68374𝑥 Question #3 For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places. x 1 2 3 4 5 6 7 8 9 10 f ( x) 20 21.6 29.2 36.4 46.6 55.7 72.6 87.1 107.2 138.1 Answer: Step 1 Remember that regression analysis is the process of looking for a best fit of model for a set of data. This can be done on a graphing utility as follows: 1. Press [STAT], the input corresponging x-values of data in L1, and y-values of data in L2. 2. Use [STATPLOT] to observe a scatterplot of the data. 3. Press [STAT], then [CALC] then [ExpReg]/[LnReg]/[Logistic]. This will show you a function in either the form of an exponential, a logarithmic or a logistic model. 4. Graph this equation on the same window as the scatterplot to see if it fits the data. Step 2 1. Press [STAT], the input corresponging x-values of data in L1, and y-values of data in L2. 2. Use [STATPLOT] to observe a scatterplot of the data. Step 3 Based on the plots of the points, it can be exponential or logarithmic. However, upon checking both regression analysis, the one with the closest value of 𝑟 2 to 1 is exponential, hence, its formula is 𝑦 = 15.10062(1.24621) 𝑥 . The graph of which is below: Answer: 𝑦 = 15.10062(1.24621) 𝑥 Question #4 Sketch a graph of the function. Use transformations of functions whenever possible. f(x)=-|x| Answer: Step 1 First graph function y=|x| Then, to obtain graph of function f(x)=-|x| , do following transformations: -reflect with respect to x-axis On graph: Red - y=|x| Blue - f(x)=-|x| Step 2 Question #5 For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places. x 3 4 5 6 7 8 9 10 11 12 13 f ( x) 13.98 17.84 20.01 22.7 24.1 26.15 27.37 28.38 29.97 31.07 31.43 Answer: Step 1 Remember that regression analysis is the process of looking for a best fit of model for a set of data. This can be done on a graphing utility as follows: 1. Press [STAT], the input corresponging x-values of data in L1, and y-values of data in L2. 2. Use [STATPLOT] to observe a scatterplot of the data. 3. Press [STAT], then [CALC] then [ExpReg]/[LnReg]/[Logistic]. This will show you a function in either the form of an exponential, a logarithmic or a logistic model. 4. Graph this equation on the same window as the scatterplot to see if it fits the data. Step 2 1. Press [STAT], the input corresponging x-values of data in L1, and y-values of data in L2. 2. Use [STATPLOT] to observe a scatterplot of the data. Step 3 Based on the plots of the points, it can be exponential or logarithmic. However, upon checking both regression analysis, the one with the closest value of 𝑟 2 to 1 is logarithmic, hence, its formula is 𝑦 = 0.89478 + 12.03885 ln(𝑥). The graph of which is below: Question #6 1 Sketch a graph of the function. Use transformations of functions whenever possible. 𝑓(𝑥) = (𝑥 − 5), 2 ≤ 3 𝑥≤8 Answer: 1 We need to graph the given function 𝑓(𝑥) = (𝑥 − 5) for the restricted domain of 2 ≤ 𝑥 ≤ 8 . Since f(x) 3 is linear, we can graph the function by finding its endpoints at x=2 and x=8 and then connecting the two points with a segment. Evaluating f(x) at x=2 and x=8 gives: 1 1 𝑓(2) = (2 − 5) = (−3) = −1 3 1 3 1 3 3 𝑓(8) = (8 − 5) = (3) = 1 To graph f(x), you can then plot the endpoints of (2,-1) and (8,1) and then connect them with a segment: Question #7 For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic. x 0.5 1 3 5 7 10 12 13 15 17 20 f ( x) 18.05 17 15.33 14.55 4.04 13.5 13.22 13.1 12.88 12.69 12.45 Answer: Step 1 In determining what model is represented by a set of points, we follows these steps: 1. Plot the points on the graph. 2. Determine whether it is one of these: a) if the points do not lie on a straight line, then it is not a linear model b) if after drawing a straight line between two points most of the points in between lie above the line, then it is a logarithmic model. c) if after drawing a straight line between two points most of the points in between lie below the line, then it is a exponential model. Step 2 For this table of values, we plot the points, then draw a line between two points: Step 3 Since most of the points, lie below the drawn line, AND is decreasing from left to right, then this set of points represent a logarithmic model. Question #8 The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Decide which, if any, are exponential, and give the exponential models for those that are. x −2 −1 0 1 2 f ( x) 0.3 0.9 2.7 8.1 24.3 g ( x) 3 1.5 0.75 0.375 0.1875 Answer: Step 1 Recall: For the exponential function 𝑓(𝑥) = 𝐴𝑏 𝑥 1. f(0)=A 2. If x increases by 1, f(x) is multiplied by b Step 2 Observe the table for f(x) x −2 −1 0 1 2 f ( x) 0.3 0.9 2.7 8.1 24.3 Step 3 Calculate the ratios 𝑓(𝑥) 𝑓(𝑥−1) to see by how much f(x) increases when x increases by 1 x −2 −1 0 1 2 f ( x) 0.3 0.9 2.7 8.1 24.3 f ( x) 3 3 3 3 f ( x − 1) Step 4 Note that there exists a constant b=3 such that condition 2 is satisfied, which means that f is exponential, 𝑓(𝑥) = 𝐴 ⋅ 3𝑥 Furthermore, 𝐴 = 𝑓(0) = 2.7, so 𝑓(𝑥) = 2.7 ⋅ 3𝑥 Step 5 Observe that table for g(x): x −2 −1 0 1 2 g ( x) 3 1.5 0.75 0.375 0.1875 Step 6 Calculate the ratios 𝑔(𝑥) 𝑔(𝑥−1) , to see by how much g(x) incresded when x increases by 1 x −2 −1 0 1 2 g ( x)3 1.5 0.75 0.375 0.1875 g ( x) 0.5 0.5 0.5 0.5 g ( x − 1) Step 7 Note that there exists a constant b=0.5 such that condition 2 is satisfied, which means that g is exponential, 𝑔(𝑥) = 𝐴 ⋅ (0.5) 𝑥 Furthermore, A=g(0)=0.75, so 𝑔(𝑥) = 0.75 ⋅ (0.5) 𝑥 Since 0.5𝑥 = (2−1 ) 𝑥 = 2−𝑥 , we can write 𝑔(𝑥) = 0.75 ⋅ 2−𝑥 Question #9 A report tells us that in 2009, there were 870 gray wolves in Idaho, but that the population declined by 19% annual rate of decrease continues. a) Find an exponential model that gives the wolf population W as a function of the time t in years since 2009. 𝑊 = 870(1 − .19)𝑡 b) It is expected that the wolf population cannot recover if there are fewer than 25 individuals. How long must this rate of decline continue for the wolf population to reach 25? Answer: Given: The number of gray wolves in Idalho in year 2009 was 870. It is given that population decreases annually at rate 19%. To find: a) To find an exponential models that gives wolves population W as a function of time t in years since 2009. b) Number of years in which population of wolves reaches to 25. Solution: We know that when population decline continuosly then population after time t years is given as: 𝑊 = 𝑃0 (1 − 𝑟)𝑡 , where W is population after time t years, 𝑃0 is initial population and r is rate of decline We have given, 𝑃0 = 870 r=19% 19 = 100 =0.19 Therefore, an exponential models that gives wolves population is: 𝑊 = 870(1 − 0.19)𝑡 = 870(0.81)𝑡 Now, we have to find t when W=25 25 = 870(0.81)𝑡 25 (0.81)𝑡 = 870 (0.81)𝑡 = 0.028 log(0.81)𝑡 = log(0.028) 𝑡 ⋅ log(0.81) = log(0.028) 𝑡 ⋅ (−0.091) = −1.55 𝑡= −1.55 −0.091 t=17.03 Therefore, number of years the population will declined to 25 is 17.03 years. Question #10 The exponential models describe the population of the indicated country, A, in millions, t years after 2006. Which countries have a decreasing population? By what percentage is the population of these countries decreasing each year? Country 1: 𝐴 = 128.3𝑒 0.007𝑡 Country 2: 𝐴 = 142.3𝑒 −0.009𝑡 Country 3: 𝐴 = 1087.7𝑒 0.018𝑡 Country 4: 𝐴 = 27.3𝑒 0.021𝑡 Answer: Consider the provided question, The exponential models describe the population of the indicated country. A, In millions, t years after 2006, We need to find which country has the greatest growth rate' By what percentage is the population of that country increasing each year. Now, we have country has the greatest growth rate, Country 1: 𝐴 = 128.3𝑒 0.007𝑡 t=1 year \therefore 𝐴 = 128.3𝑒 0.007×1 = 128.3 × 1.00702 =129.2015 129.2015−128.3 Growth rate = × 100 128.3 =0.7026% Counrty 2: 𝐴 = 142.4𝑒 −0.009𝑡 t=1 year ∴ 𝐴 = 1087.7𝑒 0.018×1 =1107.4558 1107.4558−1087.7 Growth rate = × 100 1087.7 =-0.8959% Country 3: 𝐴 = 1087.7𝑒 0.018×1 t=1 year ∴ 𝐴 = 1087.7𝑒 0.018×1 =1107.4558 1107.4558−1087.7 Growth rate = × 100 1087.7 =1.8162% Country 4: 𝐴 = 27.3𝑒 0.0211 t=1 year ∴ 𝐴 = 27.3𝑒 0.021×1 =27.8795 Growth rate = 27.8795−27.3 27.3 × 100 =2.1227% Country 4 has the greatest growth rate The population of that country is increasing by 2.1227% each year Question #11 The exponential models describe the population of the indicated country, A, in millions, t years after 2010. Use these models to solve, What was the population of Japan in 2010? India: 𝐴 = 1173.1𝑒 0.008𝑡 Iraq: 𝐴 = 31.5𝑒 0.019𝑡 Japan: 𝐴 = 127.3𝑒 −0.006𝑡 Russia: 𝐴 = 141.9𝑒 −0.005𝑡 Answer: Given: The exponential models describe the population of Japan after 2010: 𝐴 = 127.3𝑒 −0.006𝑡 Now, The time t in 2010 is: t=2010-2010=0 The population of Japan in 2010 is: 𝐴 = 127.3𝑒 −0.006𝑡 = 127.3𝑒 −0.006×0 = 127.3𝑒 0 =127.3 millions Question #12 The exponential models describe the population of the indicated country, A, in millions, t years after 2010. Use these models to solve, When will India's population be 1491 million India: 𝐴 = 1173.1𝑒 0.008𝑡 Iraq: 𝐴 = 31.5𝑒 0.019𝑡 Japan: 𝐴 = 127.3𝑒 −0.006𝑡 Russia: 𝐴 = 141.9𝑒 −0.005𝑡 Answer: Here we have four different exponential models. To calculate the population of India, we use the exponential model of India. That is 𝐴 = 1173.1𝑒 0.008𝑡 By substituting the value of A=1491, ⇒ 1491 = 1173.1𝑒 0.008𝑡 By dividing both sides by 1173.1 ⇒ 1.27099 = 𝑒 0.008𝑡 By taking natural log(𝐿𝑁) of both sides, ⇒ 0.2398 = 0.008𝑡 ⇒ 𝑡 = 29.947 ≈ 30 That is, after 30 years, the population of India would be 1491 million. That is, 2010+30=2040 Therefore, the population of India would be 1491 million in the year 2040. Question #13 The exponential models describe the population of the indicated country, A, in millions, t years after 2010. Use these models to solve, When will India’s population be 1377 million? India: 𝐴 = 1173.1𝑒 0.008𝑡 Answer: Given: Population of India, A, in millions, t years after 2010 is given by the exponential model 𝐴 = 1173.1𝑒 0.008𝑡 When A=1377 million 1377 = 1173.1𝑒 0.008𝑡 1377 = 𝑒 0.008𝑡 1173.1 1.1738 = 𝑒 0.008𝑡 ln (1.1738) = ln (𝑒 0.008𝑡 = 0.1602 = 0.008𝑡 0.1602 𝑡= 0.008 t=20.025 𝑡 ≈ 20 India’s population will be 1377 million in 20 years i.e. in year 1397. Question #14 The exponential models describe the population of the indicated country, A, in millions, t years after 2010.What was the population of Iraq in 2010? Iraq: 𝐴 = 31.5𝑒 0.019𝑡 Answer: The exponential models for the population of Iraq (in millions), t year after 2010 is given as, 𝐴 = 31.5𝑒 0.019𝑡 To find the population of Iraq in year 2010, we need to find the value of t t=2010-2010 =0 Putting t=0 in the population model of Iraq, we get 𝐴 = 31.5𝑒 0.019(0) = 31.5 ⋅ (1) =31.5 Therefore, the population of Iraq in year 2010 is 31.5 million. Question #15 The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Give the exponential models for those that are. (If an answer does not exist, enter DNE.) x −2 −1 0 1 2 f ( x) 2 4 8 16 32 g ( x) 7 0 1 0 7 Answer: Given: Values of two functions f(x) and g(x) are given. From the given data, f(x) is the only function which is exponential. Finding Equation of f(x): First term, a=2 4 Growth factor, 𝑏 = = 2 2 Equation of the exponential function is given by: ⇒ 𝑓(𝑥) = 𝑎(𝑏) 𝑥 ⇒ 𝑓(𝑥) = 2(2) 𝑥 Question #16 The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Give the exponential models for those that are. f(x)=? g(x)=? x −2 −1 0 1 2 f ( x) 1.125 2.25 4.5 9 18 g ( x) 16 8 4 2 1 Answer: Given table show the value of the two functions f(x) and g(x) at -2,-1,0,1,2. To find f(x): From the table it is clear that for every increment of x by 1 unit, the value of f(x) gets multiplied by 2. So, let 𝑓(𝑥) = 𝑘. 2𝑥 . Given 𝑓(0) = 4.5 ⇒ 𝑘. 20 = 4.5 ⇒ 𝑘 = 4.5. Hence f(x)=(4.5).2x To find g(x): From the table it is clear that for every increment of x by 1 unit, the value of f(x) get halved. 1 𝑥 So let, 𝑔(𝑥) = 𝑘. ( ) . 2 1 0 Given 𝑔(0) = 4 ⇒ 𝑘. ( ) = 4 ⇒ 𝑘 = 4. 1 𝑥 2 Hence 𝑔(𝑥) = 4. ( ) . 2 1 𝑥 Thus, the exponential modes of two function are 𝑓(𝑥) = (4.5). 2𝑥 , 𝑔(𝑥) = 4. ( ) . 2 Question #17 The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Give the exponential models for those that are. x −2 −1 0 1 2 f ( x) 31.5 10.5 3.5 10.5 31.5 g ( x) 0.2 0.6 1.8 5.4 10.8 Answer: Consider the table of f(x) x −2 −1 0 1 2 f ( x) 31.5 10.5 3.5 10.5 31.5 The difference of f(x) is given as 10.5-31.5=-20 3.5-10.5=-7 10.5-3.5=7 31.5-10.5=20 Thus, the function f(x) is not growing exponentially and thus exponential function for f(x) doesn't exist. Hence f(x)=DNE Consider the table of g(x) x −2 −1 0 1 2 g ( x) 0.2 0.6 1.8 5.4 10.8 as we can that g(x) has same multiplier as 3 common ratio 0.6 =3 0.2 1.8 =3 0.6 5.4 =3 1.8 10.8 5.4 =3 Hence common ratio b=3 and initial value is a=1.8 Hence, 𝑔(𝑥) = 1.8(3) 𝑥 Question #18 The exponential models describe the population of the indicated country A, in millions, t years after 2006. Which country has the greatest growth rate? By what percentage is the population of that country increasing each year? Country: 1 𝐴 = 127.8𝑒 0.006𝑡 Country 2: 𝐴 = 1096.9𝑒 0.018𝑡 Country 3: 𝐴 = 147.4𝑒 −0.003𝑡 Country 4: 𝐴 = 25.6𝑒 0.023𝑡 Answer: Given data: Country: 1 𝐴 = 127.8𝑒 0.006𝑡 Country 2: 𝐴 = 1096.9𝑒 0.018𝑡 Country 3: 𝐴 = 147.4𝑒 −0.003𝑡 Country 4: 𝐴 = 25.6𝑒 0.023𝑡 The highest growth rate would be the country having the largest exponent in growth function Country 4 have the largest exponent in growth function , 𝑒 0.023 Now, For growth rate, we have to divide the (t+1) to t 𝐴(𝑡+1) 𝐴(𝑡) is the ratio of one year's population to the previous year's. 𝐴(𝑡 + 1) 25.6 ⋅ 𝑒 0.023(𝑡+1) = = 1.023 𝐴(𝑡) 25.6 ⋅ 𝑒 0.023(𝑡) Country 4 population is increasing by 2.3 % each year Question #19 By 2009, the models indicate that Canada’s population will exceed Uganda’s by approximately 2.8 million.The exponential growth models describe the population of the indicated country, A, in millions, t years after 2006. Determine whether the statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Canada: 𝐴 = 33.1𝑒 0.009𝑡 Uganda: 𝐴 = 28.2𝑒 0.034𝑡 Answer: Given: Growth model for Canada 𝐴 = 33.1𝑒 0.009𝑡 . Growth model for Uganda 𝐴 = 28.2𝑒 0.034𝑡 Canada’s population- Uganda’s population= 2.8 million. As given population A in million and t years after 2006. Therefore, t=0 for 2006 than for 2009 t will be 3. Now, calculate the population of each country at 2009 For Canada population put t=3 in 𝐴 = 33.1𝑒 0.009𝑡 . 𝐴 = 33.1𝑒 0.009(3) 𝐴 = 33.1𝑒 0.027 𝐴 = 34.006 . For Uganda population put t=3 in 𝐴 = 28.2𝑒 0.034𝑡 𝐴 = 28.2𝑒 0.034(3) 𝐴 = 28.2𝑒 0.102 A=31.228 Therefore, by 2009 Canada’s population is 34.006 million, and Uganda’s population is 31.228 million. Now, Canada’s population- Uganda’s population=34.006-31.228 =2.778 Which is approximately 2.8 million. Therefore, the given statement is true. Answer: Given statement By 2009, the models indicate that Canada’s population will exceed Uganda’s by approximately 2.8 million is true. Question #20 The exponential models describe the population of the indicated country. A, in millions, t years after 2006. Which country has the greates growth rate? By what percentage is the population of that country increasing each year? Country1: 𝐴 = 129.3𝑒 0.001𝑡 Country 2: 𝐴 = 1096.9𝑒 0.011𝑡 Country 3: 𝐴 = 28.7𝑒 0.028𝑡 Country 3: 𝐴 = 147.9𝑒 −0.004𝑡 Answer: We can solve it using exponential growth rate formula as below. We have the exponential growth rate formula as Final amount A=(principal value P) × 𝑒 (rate of interes)(tome) so from the given data we can extract the rate of interest of country A=0.001 country B=0.011 country C=0.028 country D=-0.004 Here we can see the highest growth rate is 0.028, so its the 1st answer 0.028 again the percentage growth rate for this country would be = 2.8% per year 100