Questions and Answers #2 Modeling

Questions and Answers Sheet 2 Modeling Question #1 To find: The year in which the 2006 cost of tution, room and board fees in public colleges will be doubled using the function 𝑓(𝑥) = 13.017(1.05) 𝑥 Answer: The given model is 𝑓(𝑥) = 13.017(1.05) 𝑥 Where x is the number of years since 2006 and =f(x) is the cost in dollars. From the table, the average annual cost in 2006 is $12.837. After x years the 2006 will be doubled So, After x years, the cost will be 2 × 12837 = $25674 Hence, 𝑓(𝑥) = 13.017(1.05) 𝑥 = 25674 25674 (1.05) 𝑥 = 13017 Taking natural logarithm on each side ln(1.05) 𝑥 = ln 25674 13017 Using calculator, 𝑥(ln 1. 05) = 0.679 Divide by ln 1. 05, 𝑥 = 0.679 ln 1.05 Using calculator, x=13.92 Rounded off to nearest tens, 𝑥 ≈ 13 Hence, based on this model the 2006 cost will be doubled in 13 years since 2006. That is, the year = 2006 + 13 = 2019 Final statement: Hence, based on this model the 2006 cost will be doublend in 2019. Question #2 To find: The number of students took at least one online college course in 2012, using the given model, Number of students = 0.0112𝑥 2 + 0.4663𝑥 + 1.513. Answer: The given model is Number of students = 0.0112𝑥 2 + 0.4663𝑥 + 1.513 Using this model, we can approximate the number of students (in millions) taking at least one online college course between the period 2002 and 2012. Here, x=0 corresponds 2002 x=1 corresponds 2003. So, x=12 corresponds 2012. So, substituting x=12 in the given model, we get the approximate number of students (in millions) who took at least one online college course in 2012. So, Number of students in 2012 = 0.0112(10)2 + 0.4663(10) + 1.513 = 0.0112(100) + 4.663 + 1.513 = 7.296 millions Therefore, 7.296 million students took at least one online college course in 2012. Question #3 Aurora is planning to participate in an event at her school's field day that requires her to complete s at various stations in the fastest time possible. To prepare for the event, she is practicing and keeping track of her time to complete each station. The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the . (1, 3), (2, 6), (3, 12), (4, 24) Part A: Is this data modeling an algebraic sequence or a geometric sequence? Explain your answer. Part B: Use a recursive formula to determine the time she will complete station 5. Show your work. Part C: Use an explicit formula to find the time she will complete the 9th station. Show your work. Answer: Step 1 We have, Aurora is practicing and keeping track of her time to complete each station. The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the . (1, 3), (2, 6), (3, 12), (4, 24). (A) Here, y-coordinate have values as 𝑎1 = 3, 𝑎2 = 6, 𝑎3 = 12, 𝑎4 = 24, 𝑎5 ⋯ 𝑎𝑛 In the above sequence, 𝑎2 6 = =2 𝑎1 𝑎3 𝑎2 = 3 12 6 =2 Hence, above sequence data is modeling a geometric sequence because it follow the geometric sequence pattern having same common ratio. Step 2 (B) We know that for a geometric sequence, 𝑛𝑡ℎ term of sequence having first term a and common ratio r is given as 1) 𝑎𝑛 = 𝑎𝑟 𝑛−1 From the given data, x-coordinate corresponds to station number and y-coordinate corresponds to time t. For station5, we need to calculate 5𝑡ℎ term of the sequence given by equation (1). On substituting a=3, r=2, and n=5 in equation (1) and using explicit formula to determine the time she will complete station 5 as 𝑎5 = (3) ⋅ (2)5−1 = (3) ⋅ (2)4 = (3) ⋅ (16) =48 minutes Hence, the time she will complete station 5 is 48 minutes. (C) For station 9, we need to calculate 9𝑡ℎ term of the sequence given by equation (1). On substituting a=3, r=2, and n=5 in equation (1) and using explicit formula to determine the time she will complete station 9 as 𝑎9 = (3) ⋅ (2)9−1 = (3) ⋅ (2)2 = (3) ⋅ (256) =768 minutes Hence, the time she will complete station 9 is 768 minutes. Question #4 Whether the given function is an appropriate model for extended years and explain the reason. Given: The polynomial function is 𝑓(𝑥) = −0.87𝑥 3 + 0.35𝑥 2 + 81.62𝑥 + 7684.94 Here, x represents the number of years after 1987 and f(x) represents the number of thefts in that respective year. Answer: Procedure used: Leading Coefficient Test: As x increases or decreases without bound, the graph of the polynomial function. 𝑓(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 (𝑎𝑛 ≠ 0) eventually rises or falls. In particular, (1) For n odd: If the leading coefficient is positive, the graph falls to the left and rises to the right. If the leading coefficient is negative, the graph rises to the left and falls to the right. (2) For n even: If the leading coefficient is positive, the graph rises to the left and rises to the right. If the leading coefficient is negative, the graph falls to the left and falls to the right. Description: The degree of the given function is observed to be 3, which is odd. The odd polynomial function have opposite behavior at each end. The number 𝑎𝑛 , which is the coefficient of the variable with highest power is called the leading coefficient. The coefficient x^{3} is noticed to be -0.87. That is, the leading coefficient is negative. Thus, by above procedure, the graph of 𝑓(𝑥) = −0.87𝑥 3 + 0.35𝑥 2 + 81.62𝑥 + 7684.94 rises to the left and falls to the right. This implies that the number of thefts in x years after 1987 will be negative as the number the years increases. This is impossible. So, it is not capable for modeling the number of thefts in United States for the extended years. Question #5 To find: a) radiative force R when the carbon dioxide level is double the preindustrial amount of carbon dioxide in 𝐶 watts per square meter by using R=6.3 In . 𝐶0 b) the global temperature increase T by using 𝑇(𝑅) = 1.03𝑅 Answer: Concept: Modeling is a method of simulating the real life situations with mathematical equations. Using these models we can forecast the future behavior. We can translate the mathematical word problem into a math expression using variables. Calculation: 𝐶 The given model is 𝑅 = 𝑘𝐼𝑛 , R is the radiative forcing in watts per square meter. 𝐶0 Where C_{0} is the preindustrial amount of carbon dioxide, C is the current level of carbon dioxide. And, according to IPCC, k=6.3 𝐶 a) Let 𝑅 = 6.3𝐼𝑛 𝐶0 And given that, the carbon dioxide level = double the preindustrial amount of carbon dioxide That is, 𝐶 = 2𝐶0 So, let 𝐶 = 2𝐶0 R=6.3 In 2𝐶0 𝐶0 Cancel 𝐶0 , 𝑅 = 6.3𝐼𝑛 2𝐶0 𝐶0 R=6.3 In 2 Using calculator, 𝑅 = 6.3 × 0.6931 R=4.3668 𝑅 ≈ 4.37 watts per square meter. Hence, radiative force R when the carbon dioxide level is double the preindustrial amount of carbon dioxide is 4.37 watts per square meter. b) Let R=4.37 watts per square meter And the global temperature increase T(R)=1.03R So, when 𝑅 = 4.37, 𝑇(𝑅) = 1.03 × 4.37 Therefore, T(R)=4.50 Final statement: Based on the function, a) radiative force R when the carbon dioxide level is double the preindustrial amount of carbon dioxide is 4.37 watts per square meter. b) the global temperature increase T=4.50 . Question #6 To find: The approximated quarterback rating (to the nearest tenth) of Drew Brees of the New Orleans Saints during the 2013 regular season using, Rating = 𝐶 𝐴 𝑇 𝐴 𝑌 𝐴 𝐼 𝐴 (250⋅ )+(1000⋅ )+(12.5⋅ )+6.25−(1250⋅ ) 3 . Answer: Concept: Modeling is a method of simulating the real life situations with mathematical equations. Using these models we can forecast the future behavior. We can translate the mathematical word problem into a math expression using variables. Calculation: The given model is Rating = 𝐶 𝐴 𝑇 𝐴 𝑌 𝐴 𝐼 𝐴 (250⋅ )+(1000⋅ )+(12.5⋅ )+6.25−(1250⋅ ) 3 . Using this model, we can approximate the quarterback rating (to the nearest tenth) of Drew Brees of the New Orleans Saints during the 2013 regular season. He attempted 650 passes, completed 446, had 5162 total yards, threw for 39 touchdowns, and had 12 interceptions. Here, A= attempted passes =650 , C= completed passes =446 , T= touchdown passes =39 , Y= yards gained passing =5162 , and I= interceptions =12 . [Weighting factors] the four category ratios are limited to nonnegative values with the following maximums. 𝐶 𝑇 𝑌 𝐼 = 0.775, = 0.11875, = 12.5 and = 0.095 𝐴 𝐴 𝐴 𝐴 Substituting the values in given model, we get 𝐶 𝑇 𝑌 𝐼 (250 ⋅ ) + (1000 ⋅ ) + (12.5 ⋅ ) + 6.25 − (1250 ⋅ ) 𝐴 𝐴 𝐴 𝐴 = 3 = (250⋅ 446 39 5162 12 )+(1000⋅ )+(12.5⋅ )+6.25−(1250⋅ ) 650 650 650 650 3 (250 ⋅ 0.686) + (1000 ⋅ 0.06) + (12.5 ⋅ 7.941) + 6.25 − (1250 ⋅ 0.018) = 3 171.5+60+99.262+6.25−22.5 = 3 = 314.512 3 =104.83 Final statement: Therefore, the approximated quarterback rating (to the nearest tenth) of Drew Brees of the New Orleans Saints during the 2013 regular season is 104.83. Question #7 A person is making a 5-gallon punch for a party that will consist of two different types of juices; apple and orange juice. They use at least 1 gallon of each type but want to use more apple than orange juice. Given that x represents the gallons of apple juice in the punch and y represents the gallons of orange juice in the punch, then create a system of equations and/or inequalities modeling the scenario. Select all that apply. a) x+y=5 b) x-y=5 c) x>y d) x𝑦 option (3) is correct. Question #8 To solve: The equation E=IZ for given 𝐸 = 57 + 67𝑖, 𝑍 = 9 + 5𝑖 Answer: 1) Concept: Mathematical modeling is the application of the mathematical concepts in various field of study and also in daily practical side of finding solutions for various problems. In Ohm's law E=IZ any of the unknown value can be found provided the value of the others two. i.e. E=IZ 𝐸 𝐼= 𝑍= 𝑍 𝐸 𝐼 2) Calculation: Given, E=57+67i, Z=9+5i E=IZ Dividing both sides on I 𝐸 𝐼𝑍 = 𝐼 𝐸 𝑍 𝑍 =𝐼 𝐼= 𝐸 𝑍 On substituting the values of E and Z 𝐼= 57+67𝑖 9+5𝑖 Multiplying the conjugate 9-5i by the denominator with both the numerator and with the denominator = = 57+57𝑖 × 9−5𝑖 9+5𝑖 9−5𝑖 (57+67𝑖)(9−5𝑖) (9+5𝑖)(9−5𝑖) Multiply each term of the first factor with each term of the factor in the numerator. = = 57⋅9+57⋅(−5𝑖)+(67𝑖)⋅9+(67𝑖)⋅(−5𝑖) (9+5𝑖)(9−5𝑖) 513−285𝑖+603𝑖−335𝑖 2 (9+5𝑖)(9−5𝑖) Since 𝑖 2 = −1 = = 513−285𝑖+603𝑖−335(−1) (9+5𝑖)(9−5𝑖) 513−285𝑖+603𝑖+335 (9+5𝑖)(9−5𝑖) Adding the real parts 513 and 335 together and the imaginary parts -285i and 603i together in the numerator = (513+335)+(−285𝑖+603𝑖) (9+5𝑖)(9−5𝑖) Applying the rule for the product of sum and difference of two terms (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2 = = 848+318𝑖 92 −(5𝑖)2 848+318𝑖 81−25𝑖 2 Since i^{2}=-1 = = = = 848+318𝑖 81−25(−1) 848+318𝑖 81+25 848+318𝑖 106 8⋅106+3⋅106 106 = 8⋅106 106 + 3⋅106 106 𝐼 = 8 + 3𝑖 Final Statement: Therefore, the ohm's law current for the given value of voltage and impedance is 8+3i Question #9 To find: The total sales T(x) of Northwest Creations, to the nearest dollar using the model where the distribution of x thousand catalogues is approximated by 𝑇(𝑥) = 5000 log(𝑥 + 1). Answer: Concept: Modeling is a method of simulating the real life situations with mathematical equations. Using these models we can forecast the future behavior. We can translate the mathematical word problem into a math expression using variables. Calculation: The given model is 𝑇(𝑥) = 5000 log(𝑥 + 1) Where x is the number of catalogues distributed in throusands and T(x) is the total sales in dollars. So, in order to get the total sales T(x), replace x in the given model with the number of catalogues. a) 𝑇(𝑥) = 5000 log(𝑥 + 1) Replace x with 5000, 𝑇(5000) = 5000 log(5000 + 1) Since log 𝑥 = log10 𝑥, 𝑇(5000) = 5000 log10 5 001 Using calculator, 𝑇(5000) ≈ 5000 × 3.699 𝑇(5000) ≈ 18495 Based on this model, the total sales of the Northwest Creations resulting from 5000 catalogues is $18495 b) 𝑇(𝑥) = 5000 log(𝑥 + 1) Replace x with 24000, 𝑇(24000) = 5000 log(24000 + 1) Since log 𝑥 = log10 𝑥 , 𝑇(24000) = 5000 log10 2 4001 Using calculator, 𝑇(24000) ≈ 5000 × 4.3802 𝑇(24000) ≈ 21901 Based on this model, the total sales of the Northwest Creations resulting from the distribution of 24000 catalogues is $21901 c) 𝑇(𝑥) = 5000 log(𝑥 + 1) Replace x with 49000, 𝑇(49000) = 5000 log(49000 + 1) Since log 𝑥 = log10 𝑥 , we have 𝑇(49000) = 5000 log10 4 9001 Using calculator, 𝑇(49000) ≈ 5000 × 4.6902 𝑇(49000) ≈ 23451 Based on this model, the total sales of the Northwest Creations resulting from 49000 catalogues is $23451 Final statement: Based on this model, the total sales of the Northwest Creations resulting from 5000 catalogues is $18495, from 24000 catalogues is $21901 and from 49000 catalogues is $23451. Question #10 To solve: The equation E=IZ for given I=10+4i, E=88+128i Answer: 1) Concept: Mathematical modeling is the application of the mathematical concepts in various field of study and also in daily practical side of finding solutions for various problems. In Ohm's law E=IZ any of the unknown value can be found provided the value of the others two. i.e. E=IZ 𝐸 𝐼= 𝑍 𝑍= 𝐸 𝐼 2) Calculation: Given, 𝐼 = 10 + 4𝑖, 𝐸 = 88 + 128𝑖 𝐸 = 𝐼𝑍 Dividing both sides on I 𝐸 𝐼𝑍 = 𝐼 𝑍 𝐸 =𝑍 𝐼 𝑍= 𝐸 𝐼 On substituting the values of E and I 88 + 128𝑖 𝑍= 10 + 4𝑖 2(44+64𝑖) = = 2(5+2𝑖) 44+64𝑖 5+2𝑖 Multiplying the conjugate 5-2i by the denominator with both the numerator and with the denominator = = 44+64𝑖 × 5−2𝑖 5+2𝑖 5−2𝑖 (44+64𝑖)(5−2𝑖) (5+2𝑖)(5−2𝑖) Multiply each term of the first factor with each term of the factor in the numerator. = = = = 44⋅5+44⋅(−2𝑖)+64𝑖⋅5+64𝑖⋅(−2𝑖) (5+2𝑖)(5−2𝑖) 220−88𝑖+320𝑖−128𝑖 2 (5+2𝑖)(5−2𝑖) 220−88𝑖+320𝑖−128(−1) (5+2𝑖)(5−2𝑖) 220−88𝑖+320𝑖+128 (5+2𝑖)(5−2𝑖) Combining the real and imaginary parts = (220+128)+(320𝑖−88𝑖) (5+2𝑖)(5−2𝑖) 348+232𝑖 = (5+2𝑖)(5−2𝑖) Applying the rule for the product of sum and difference of two terms (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 + 𝑏 2 348+232𝑖 = (5)2 = −(2𝑖)2 348+232𝑖 25−4𝑖 2 Since 𝑖 2 = −1 348 + 232𝑖 = 25 − 4(−1) 348+232𝑖 = = 25+4 348+232𝑖 29 Dividing each term of the numerator by 29 348 232 = += ; 29 29 29⋅12 8⋅29 = + 29 29 𝑍 = 12 + 8𝑖 Final Statement: Therefore, the ohm's law voltage for the given value of voltage and current and is 12+8i.