Questions and Answers #3 Modeling

Questions and Answers Sheet 3 Modeling Question #1 A student bought a calculator and a textbook for a course in algebra. He told his friend that the total cost was $145(without tax) and that the calculator cost $20 more than four times the cost of the textbook. What was the cost of each item? Let x= the cost of a calculator and y= the cost of the textbook. The  x + y = 145  x = 4 y + 20 corresponding modeling system is  Solve the system by using the method of substitution and enter the ordered pair. Answer: Step 1 To solve the system of equations by using method of substitution Step 2 Given: 𝑥 + 𝑦 = 145(1) 𝑥 = 4𝑦 + 20(2) Substitute the value of x from equation 2 in 1 (4𝑦 + 20) + 𝑦 = 145 ⇒ 5𝑦 = 145 − 20 ⇒ 5𝑦 = 125 ⇒ 𝑦 = 125/5 = 25 ∴ 𝑥 = 4 × 25 + 20 = 120 Hence, (𝑥, 𝑦) = (120,25) Question #2 Twelve gallons of a salt solution consists of 35% salt. It is the result of mixing a 55% solution with a 25% solution. How many gallons of each of the solutions was used? Let x= the number of gallons of the 55% solution and y= the number of gallons of the 25% solution. The corresponding modeling system is  x + y = 12  0.55 x + 0.25 y = 0.35(12) Solve the system by using the method of substitution and what is the ordered pair? Answer: Step 1 Given system is 𝑥 + 𝑦 = 12(1) 0.55𝑥 + 0.25𝑦 = 0.35(12)(2) We solve this system using substitution Step 2 From (1)𝑦 = 12 − 𝑥(3) We substitute y=12-x in (2) we get 0.55𝑥 + 0.25(12 − 𝑥) = 0.35𝑥(12) ⇒ 0.55𝑥 + 3 − 0.25𝑥 = 4.2 ⇒ 0.3𝑥 = 4.2 − 3 ⇒ 0.3𝑥 = 1.2 12 1.2 10 12 10 ⇒𝑥= = = × =4 3 0.3 10 3 10 From (3) 𝑦 = 12 − 4 = 8 Hence the solution (𝑥, 𝑦) = (4,8) Question #3 To find: a) The speed of the sprinter while he running as crossed the finish line b) After how many seconds was he running at the rate of 10 m per sec 𝑡 Using the model function 𝑓(𝑡) = 11.65 (1 − 𝑒 −1.27 ) where f(t) is the speed of the sprinter per second after t seconds. 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: 𝑡 The given model function is 𝑓(𝑡) = 11.65 (1 − 𝑒 −1.27 ) where f(t) is the speed of the sprinter per second after t seconds. a) The sprinter finished the race in 9.86 seconds. So, substituting x=9.86 , we het the speed of the sprinter while he crossing the finish line. 9.86 Let 𝑥 = 9.86, 𝑓(9.86) = 11.65 (1 − 𝑒 −1.27 ) Using the calculator, 𝑓(9.86) = 11.65 × (1 − 0.00042484781) 𝑓(9.86) = 11.65 meters per second if rounded to nearest hundredth. The speed of the sprinter while he running as crossed the finish line is 11.645 m/sec. b) The speed of the sprinter is given by 10 m/sec. So, f(t)=10 𝑡 So, 𝑓(𝑡) = 11.65 (1 − 𝑒 −1.27 ) = 10 𝑡 Divide each side by 11.65, (1 − 𝑒 −1.27 ) = 10 11.65 𝑡 Using calculator, (1 − 𝑒 −1.27 ) = 0.858369 𝑡 1 − 0.858369 = 𝑒 −1.27 𝑡 𝑒 −1.27 = 0.1416 𝑡 Taking natural logarithm on each side, In (𝑒 −1.27 ) = 𝐼𝑛0.1416 Using calculator, (− − 𝑡 1.27 𝑡 1.27 ) log 𝑒 𝑒 = log 𝑒 0. 1416 = −1.9547 Divide by −1, 𝑡 1.27 = 1.9547 Using calculator, 𝑡 = 1.9547 × 1.27 Rounded to nearest hundredth, 𝑡 ≈ 2.48 seconds. So, after 2.48 seconds his speed was 10 m/sec. Final statement: a) The speed of the sprinter while he running as crossed the finish line is 11.65 m/sec. b) After 2.48 seconds was he running at the rate of 10 m per sec. Question #4 To find: The percent of moisture in the central Sierra Nevada (a mountain range in California), to the nearest tenth that falls as snow at each altitude (a) 3000 ft (b) 4000 ft (c) 7000 ft using the model f(x)=86.3 In x-680 , where x is altitude in feet and f(x) is the percent of moisture that falls as snow. 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 f(x)=86.3 In x-680 where x is altitude in feet and f(x) is the percent of moisture that falls as snow. So, in order to get the percent of moisture f(x), replace x in the given model with the values of altitudes. a) Let x=3000ft f(x)=86.3 In x-680 Replace x with 3000, 𝑓(3000) = 86.3(𝐼𝑛3000) − 680 Using calculator, 𝑓(3000) ≈ 86.3 × 8.00636 − 680 Multiplying, 𝑓(3000) ≈ 690.9495 − 680 Subtracting, 𝑓(3000) ≈ 10.9495 Rounded to nearest tenth, 𝑓(3000) ≈ 10.95 Based on this model, 10.95% of moisture falls as snow. b) Lex x=4000ft f(x)=86.3 In x-680 Replace x with 3000, 𝑓(4000) = 86.3(𝐼𝑛4000) − 680 Using calculator, 𝑓(4000) ≈ 86.3 × 8.294049 − 680 Multiplying, 𝑓(4000) ≈ 717.776483 − 680 Subtracting, 𝑓(4000) ≈ 35.7764 Rounded to nearest tenth, 𝑓(4000) ≈ 35.78 Based on this model, 35.78% of moisture falls as snow. c) Let x=7000ft f(x)=86.3 In x-680 Replace x with 7000, f(7000)=86.3 (In 7000)-680 Using calculator, 𝑓(7000) ≈ 86.3 × 8.853665 − 680 Multiplying, 𝑓(7000) ≈ 764.071326 − 680 Subtracting, 𝑓(7000) ≈ 84.07132 Rounded to nearest tenth, 𝑓(7000) ≈ 84.07 Based on this model, 84.07% of moisture falls as snow. Final statement: Based on this model, 10.95% of moisture falls as snow at 300 ft, 35.78% of moisture falls as snow at 4000 ft and 84.07% of moisture falls as snow at 7000 ft. Question #5 To find: The points where the comet might intersect the orbiting planet. It is given that the planet's orbit follows a path described by 16𝑥 2 + 4𝑦 2 = 64. Given that a comet follows the parabolic path 𝑦 = 𝑥 2 − 4. Answer: Procedure used: An Application of a Nonlinear System: Step 1: Write a system of equations modeling the conditions in the problem. Step 2: Solve the system and answer the question asked in the problem. Step 3: Check the proposed solution in the original wording of the problem. Calculation: Multiply 𝑦 = 𝑥 2 − 4 by 16 and obtain the equation 16𝑦 = 16𝑥 2 − 64. Add the both equations and obtain the result as follows. (16𝑥 2 + 4𝑦 2 ) + (16𝑦) = 64 + (16𝑥 2 − 64) 4𝑦 2 + 16𝑦 = 0 𝑦 2 + 4𝑦 = 0 𝑦(𝑦 + 4) = 0 On further simplification gives, y=0 or y+4=0 y=0 or y=-4 Substitute y=0 in the equation 𝑦 = 𝑥 2 − 4 and obtain the value of x as follows. (0) = 𝑥 2 − 4 𝑥2 = 4 𝑥 = ±2 Substitute y=-4 in the equation 𝑦 = 𝑥 2 − 4 and obtain the value of x as follows. (−4) = 𝑥 2 − 4 𝑥2 = 0 x=0 Thus, for 𝑦 = 0, 𝑥 = ±2 and for y=-4, x=0 . Thus, the solution set is {(2,0), (−2,0), (0, −4)} Check the result, by substituting the obtained solutions in the given original equations 16𝑥 2 + 4𝑦 2 = 64 and 𝑦 = 𝑥 2 − 4. Substitute (2.0) in the given system and check. 16(2)2 + 4(0)2 = 64 64=64 (0) = (2)2 − 4 0=0 Substitute (-2,0) in the given system and check. 16(−2)2 + 4(0)2 = 64 64=64 (0) = (−2)2 − 4 0=0 Substitute (0,-4) in the given system and check. 16(0)2 + 4(−4)2 = 64 64=64 (−4) = (0)2 − 4 -4=-4 Therefore, the points where the comet might intersect the orbiting planet are {(2,0), (−2,0), (0, −4)}. Question #6 To solve: The problem by modeling and solving a system of linear equation. The given problem is a mixer problem where 60% sugar solution and 30% sugar solution are mixed together to make 51% sugar solution where quantity of 30% sugar solution is to be calculated. Answer: Procedure used: ''Model and solve system of linear equations'' 1) Identify the category of given problem. 2) Recognize and algebraically name the unknowns. 3) Translate the problem statement as a system of linear equations using the variables from Step 2. 4) Solve the system as obtained in Step 3. 5) State the answer statemen.'' Calculation: Step 1: The given problem is a mixer problem where 60% sugar solution and 30% sugar solution are mixed together to make 51% sugar solution where quantity of 30% sugar solution is to be calculated. Step 2: There are two unknowns in the problem. Let total amount of 60% sugar solution is a and that of 30% sugar solution is s. Step 3: The mixer problem is summarized in Table 1. Name 60%sugar solution 30%sugar solution Amount (inliters ) Concentration a 60% s 30% Table 1 Total amount of 51% sugar solution is 10 quarts. a+s=10 In the question it is given that 60% sugar solution and 30% sugar solution are mixed together to make 51% sugar solution. 0.60a+0.30s=0.51(10) 0.60a+0.30s=5.1 Thus, the system of linear equations is: a+s=10(1) 0.60a+0.30s=5.1(2) Step 4: Substitute the value of s from equation (1) to (2). 0.60a+0.30s=5.1 0.60a+0.30(10-a)=5.1 0.60a+3-0.30a=5.1 0.30a=5.1-3 Solve the equation above to obtain the value of variable p as: 0.30a=2.1 2.1 𝑎= 0.30 a=7 Substitute the value of the variable a in equation (1) to obtain the value of the variable s as follows: (7)+s=10 s=10-7 s=3 Thus, the values of unknowns defined in Step 2 are a=7 and s=3 . Step 5: To check the solution obtained in Step 4, the total pounds are 7+3=10 and the amount of 51% sugar solution is 10 quarts 0.60(7) + 0.30(3) = 0.51(10). Step 6: Hence, the amount of 30% sugar solution is 3 quarts. Question #7 a) To calculate: A model of the form 𝑁(𝑡) = 𝑎 + 𝑏 𝑙𝑛 for the provided data. b) To calculate: The sales in a 7 weeks. c) The validity of the statement with explanation. Answer: a) Given: The sales of a book tend to increase over the short-term as word-of-mouth makes the book ''catch on.'' The number of bokks sold N(t) for a new novel t weeks after release at a certain book store is given in the table for the first 6 weeks. Weeks t 1 2 3 4 5 6 Number sold N(t) 20 27 31 35 38 39 Calculation: Consider the provided table: Weeks t 1 2 3 4 5 6 Number sold N(t) 20 27 31 35 38 39 Use graphic utility calculator Ti-83 for modeling an equation of provided data: STEP1: Press STAT key then press 1 in Ti-83 calculator. A table will arise. Fill above values of weeks and corresponding number of books sold in that table. STEP2: After filling the above data in table press CALC key and then press 0 to select ExpReg to calculate the model function of the provided data. The regression equation is, 𝑦 = 19.68 + 10.92 𝑙𝑛 The model for the provided data is, 𝑁(𝑡) = 19.68 + 10.92 ln , b) Given: The number of weeks is, 7. Formula used: 𝑁(𝑡) = 19.68 + 10.92 𝑙𝑛 Calculation: Consider the model obtained in part (a) 𝑁(𝑡) = 19.68 + 10.92 Substitute the value t=7 in the equation. 𝑁(7) = 19.68 + 10.92 ln(7) ≈ 41 The number of books sold for a week 7 would be approximately 41. c) Given: The statement: ''Is it reasonable to assume that this logarithmic trend will continue.'' No, the statement is not reasonable because the above logarithmic curve is formed due to sale of book through word of mouth method of publicity which is valid for short term only. Suppose that negative publicity of book started then the sale of book may go down may not follow logarithmic trend. Answer: No, the provided statement is not valid. Question #8 a) A model of the provided data in the form 𝑦 = 𝑎𝑏 𝑥 . b) The function obtained in part (a) as an exponent function with base e. c) To calculate: The monthly cost for January in the year 2015. Answer: a) Given: The monthly costs for a small company to do business has been increasing over time due to inflation. The below table gives the monthly cost y(in $) for the month of January for selected years. The variable t represents the number of years since 2008. Year (t = 0 is 2008) Monthly costs ($) y 0 12, 000 1 12, 400 2 12,800 3 13,300 Explanation: Consider the provided table below: Year (t = 0 is 2008) Monthly costs ($) y 0 12, 000 1 12, 400 2 12,800 3 13,300 Use graphic utility calculator Ti-83 for modeling an equation of provided data: Step1: Press STAT key then press 1 in Ti-83 calculator. A table will arise. Fill above values of years and monthly cost in that table. Step2: After filling the above data in table press CALC key and then press 0 to select ExpReg to calculate the model function of the provided data. The regression equation is 𝑦 = 11988(1.034)𝑡 b) Given: The function obtained in part (a) is: 𝑦 = 11988(1.034)𝑡 Formula used: Calculation: Consider the function obtained in part (a): 𝑦 = 11988(1.034)𝑡 Apply the property that 𝑒 (ln 𝑏)𝑡 = 𝑏 𝑡 𝑦 = 11988(ln 1.034)𝑡 𝑦 = 11988𝑒 0.034𝑡 The equivalent function of 𝑦 = 11988(1.034)𝑡 using base e is 𝑦 = 11988𝑒 0.034𝑡 c) Given: The function obtained in part (a) is, 𝑦 = 11988(1.034)𝑡 and the year 2015. Calculation: Consider the provided function: 𝑦 = 11988(1.034)𝑡 Substitute the value t=7 . 𝑦 = 11988(1.034)7 ≈ 15,150 The monthly cost for January in the year 2015 would be approximately $15,150. Question #9 Modeling Luxury Home Sales The following quadratic function in general form, 𝑆(𝑡) = 5.8𝑡 2 − 81.2𝑡 + 1200models the number of luxury home sales, S(t), in a major Canadian urban area, according to statistical data gathered over a 12 year period. Luxury home sales are defined in this market as sales of properties worth over $3 Million (inflation adjusted). In this case, t=0 represents 2000 and t=11 represents 2011. Use a calculator to find the year when the smallest number of luxury home sales occurred. Without sketching the function, interpret the meaning of this function, on the given practical domain, in one well-expressed sentence. Answer: Step 1 Given: The number of luxury home sales S(t) in a major Canadian urban arca over a period of 12 year is given by: ⇒ 𝑆(𝑡) = 5.8𝑡 2 − 81.2𝑡 + 1200 Step 2 For minimum number of sales: ⇒ ⇒ 𝑑𝑆(𝑡) =0 𝑑𝑡 𝑑(5.8𝑡 2 −81.2𝑡+1200) 𝑑𝑡 =0 ⇒ 11.6𝑡 − 81.2 = 0 ⇒ 11.6𝑡 = 81.2 81.2 ⇒𝑡= 11.6 ⇒𝑡=7 Step 3 So, minimum number of sales is given by ⇒ 𝑆(7) = 5.8(7)2 − 81.2(7) − 81.2(7) + 1200 ⇒ 𝑆(7) = 284.2 − 568.4 + 1200 ⇒ 𝑆(7) = 915.8 ⇒ 𝑆(7) ≅ 916 Question #10 There are 430 people in a wave pool. Write an inequality that represents how many more people can enter the pool. Given: Current number of people in the pool =430 Maximum capacity of pool =600 Answer: Calculation: Let x denotes number of people that can still enter the pool. Then the given situation can be written in inequality form as, 𝑥 + 430 ≤ 600 𝑥 ≤ 600 − 430 𝑥 ≤ 170