Math 115 Practice for Exam 1 Generated September 7, 2017 Name: SOLUTIONS Instructor: Section Number: 1. This exam has questions. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck. 2. Do not separate the pages of the exam. If any pages do become separated, write your name on them and point them out to your instructor when you hand in the exam.. Please read the instructions for each individual exercise carefully. One of the skills being tested on this exam is your ability to interpret questions, so instructors will not answer questions about exam problems during the exam.. Show an appropriate amount of work (including appropriate explanation) for each exercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate. 5. You may use any calculator except a TI-92 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 5 note card. 6. If you use graphs or tables to obtain an answer, be certain to include an explanation and sketch of the graph, and to write out the entries of the table that you use. 7. You must use the methods learned in this course to solve all problems. Semester Exam Problem Name Points Score Winter 2015 1 8 squirrel chase 2 10 Winter 2006 1 6 fresco 11 Fall 2015 1 7 chocolate 12 Fall 201 1 1 foot size 1 Total 7 Recommended time (based on points): 2 minutes
Math 115 / Exam 1 (February 10, 2015) page 8 8. [10 points] Throughout this page, give all answers in exact form. Do not use decimal approximations. For example, x = 1 is an exact solution to x = 1, but x = 0. is not. Sebastian has rented a helicopter to catch up to his friend Erin who is currently chasing a suspected criminal named Elphaba. When Sebastian first sees the pair they are 180 meters apart. After minutes, Erin has moved 60 meters closer to Elphaba. (In other words, the distance between them has decreased by 60 meters.) Let D(t) be the distance between Elphaba and Erin, in meters, t minutes after Sebastian begins watching them. a. [2 points] Sebastian initially assumes that D(t) is a linear function. Find a formula for D(t) under this assumption, valid for as long as it takes for Erin to catch Elphaba. Solution: If D(t) is linear, then D(t) = b+mt where b is the initial value and m is the constant averagerateofchange. Weimmediatelyknowb = 180sincethisisthe initialvalue. Wethen know the function decreases by 60 m in minutes so that s an average rate of decrease of 20m/min. Therefore D(t) = 180 20t. Answer: D(t) = 180 20t b. [1 point] After Sebastian has been watching for 6 minutes, the distance between Erin and Elphaba is 80 meters. Briefly explain why this contradicts Sebastian s initial assumption. Solution: One way to see is that the point (6,80) doesn t satisfy the formula we found in a. Another explanation would be that the average rate of change between (0,180) and (,120) is different from the average rate of change between (,120) and (6,80). c. [ points] Sebastian then determines that D(t) must in fact be an exponential function. Write a new formula for D(t) given this new information (including the data from part (b)). Remember to show your work carefully and use exact form. Solution: We now know D(t) = bc t where b is again the initial value, so D(t) = 180c t. We can solve for c by using the fact that D() = 120. Then 120 = 180c so c = ( 120 180 )1/ = ( 2 1/. ) Answer: D(t) = 180(( 120 180 ) )1 t = 180 ( ) 2 t/ d. [ points] Erin can catch Elphaba when she is within one meter of her (since Erin can jump and tackle Elphaba at this distance). Use algebra and your formula from part (c) to find how long it takes for the distance between Erin and Elphaba to decrease to 1 meter. Solution: We need to solve for t in the equation D(t) = 1. Using our formula from part (c), we have 180(( 2 ) t )1 = 1. Solving, we find (( 2 ) t )1 = 1 180 so tln((2 ) )1 = ln( 1 180 ). Finally t = ln( 1 180 ) = ln(180) ln(( 2 )1 ) ln(2) ln() = ln(180) ln(180) ln(1.5). So it takes ln(1.5) (about 8.) minutes for Erin to get within 1 meter of Elphaba. Answer: ln ( 1 180 ln( (2 ) )1 ) = ln(180) ln(1.5) Winter, 2015 Math 115 Exam 1 Problem 8 (squirrel chase 2) Solution
6 6. (11 points) A fresco supposedly painted by the Italian Renaissance artist Alessandro Botticelli (15-1510) currently contains 92% of its carbon-1 (half-life 570 years.) From this information, decide whether Botticelli could have painted the fresco. Show step-by-step calculations, and briefly explain your conclusion. 1 2 C 0 = C 0 b 570, which means: ( ) 1 1/570 = b, or b 0.999879 2 0.92 C 0 = C 0 0.999879 t, which means: ln(0.92) ln(0.999879) = t, or t 689.29 years. So, about 689.29 years have passed since the fresco was done, or the painting was done either in the year 116 or 117 (since 2006 689.29 = 116.71.) This is before Botticelli was born, so he could have not painted the fresco. Winter, 2006 Math 115 Exam 1 Problem 6 (fresco) Solution
Math 115 / Exam 1 (October 1, 2015) page 8 7. [12 points] Phillip Asafy and Genevieve Omicks both enjoy hot chocolate when it s cool outside. They made a few measurements, and these appear in the table below. P (respectively G) is Phil s (respectively Gen s) consumption of hot chocolate (in quarts, measured to the nearest tenth of a quart) in a month when the average daily high temperature is H (in degrees Celsius, measured to the nearest degree). H( C) P (quarts) G (quarts) 16.1 1. 7 12.8 11.6 15 8.0 6.5 a. [8 points] Based on this data, could either student s monthly hot chocolate consumption be reasonably modeled as a linear function of average daily high temperature? An exponential function? Neither? Carefully justify your answer in the space below. (Hint: At least one of these can be modeled by a linear or an exponential function!) Solution: First consider Phil s hot chocolate consumption. Suppose P = p(h). To check whether p(h) could be modeled by a linear function, we compute the average rate of change of p over the intervals [,7] and [7,15]. We have p(7) p() 7 = 12.8 16.1 = 0.825 and p(15) p(7) 15 7 = 8.0 12.8 8 = 0.6. Since these two average rates of change are quite different, Phil s hot chocolate consumption is not reasonably modeled by a linear function. To check whether p(h) could be modeled by an exponential function, we compute the percent rate of change of p(h) over the intervals [,7] and [7,15]. We have ( ) 1 ( p(7) 7 12.8 = p() 16.1 )1 0.9 and ( p(15) p(7) ) 1 ( )1 15 7 8.0 8 = 0.929. 12.8 The difference between these percent rates of change is less than 0.2%, so based on this data, p(h) can be reasonably modeled by an exponential function. In particular, we can check that we obtain the data in the table for P using, for example, 19.2(0.95) H Now consider Gen s hot chocolate consumption. Suppose G = g(h). From the calculations g(7) g() 7 = 11.6 1. = 0.25 and g(15) g(7) 15 7 = 6.5 11.6 8 = 0.675 we conclude that Gen s hot chocolate consumption is not reasonably modeled by a linear function. From the calculations ( ) 1 ( g(7) 7 11.6 = g() 1. )1 0.966 and ( g(15) g(7) ) 1 ( )1 15 7 6.5 8 = 0.902 11.6 we conclude that Gen s hot chocolate consumption can t be reasonably modeled by an exponential function. ( 2 (Note that for the exponential cases we could instead compare, for example, p(7) p()) with p(15) p(7).) Answers: Circle one choice for each student. Phil s consumption P: linear exponential neither linear nor exponential Gen s consumption G: linear exponential neither linear nor exponential Fall, 2015 Math 115 Exam 1 Problem 7 (chocolate) Solution
Math 115 / Exam 1 (October 1, 2015) page 9 b. [ points] For this investigation, their friend Maddy measures temperature in degrees Fahrenheit, and she measures her hot chocolate consumption in cups. She finds a function M(f) which is the number of cups of hot chocolate she consumes in a month when the average daily high temperature is f degrees Fahrenheit. If Q(H) is the number of quarts of hot chocolate Maddy consumes when the average monthly temperature is H degrees Celsius, write a formula for Q(H) in terms of M and H. Recall that there are cups in a quart and that the conversion from Fahrenheit to Celsius is given by y = 5 9 (x 2) (where y C and x F describe the same temperature). Solution: H degreescelsiusisthesameas 9 5 H+2degreesFahrenheit,andM ( 9 5 H +2 ) gives Maddy s hot chocolate consumption in cups. We divide this quantity by to convert from cups to quarts. M ( 9 5 H +2) Answer: Q(H) = Fall, 2015 Math 115 Exam 1 Problem 7 (chocolate) Solution
Math 115 / Exam 1 (October 8, 201) page 2 1. [1 points] Carla is trying to model the growth of the feet of her son, Taser, to predict what size boots she needs to buy him to last him through the winter. She has measured Taser s feet three times, once exactly nine months ago, once exactly three months ago, and once just today. Carla decides to measure t in months since she took her first measurement. Below is a table containing her measurements. Carla lost the record of her first measurement so the corresponding entry in the table is blank. t (months) 0 6 9 foot length (inches) 6. 7.2 a. [ points] Write a linear function L(t) modeling the length of Taser s feet t months after she took her first measurement. 0.267. Using point- Solution: The slope of L(t) is L(9) L(6) 9 6 slope form gives = 7.2 6. 9 6 = 0.8 L(t) = 0.8 0.8 (t 6)+6. = t+.8 b. [5 points] Write a exponential function E(t) modeling the length of Taser s feet t months after she took her first measurement. Solution: An exponential function E(t) will be of the form E(t) = ab t. Plugging in the data points gives 6. = ab 6 7.2 = ab 9. Dividing the second equation by the first gives b = 7.2 6. so b = the first equation above, we have ( (7.2 ) ) 1/ 6 6. = a 6. so a = ( 6. ) 7.2 2 5.057. Therefore our function is E(t) = 5.057(1.00) t. 6. ( ) 7.2 1/ 1.00. Using 6. Fall, 201 Math 115 Exam 1 Problem 1 (foot size) Solution
Math 115 / Exam 1 (October 8, 201) page c. [2 points] According to the exponential model you found in (b), what is the missing value in the table above? Solution: E(0) 5.057 d. [ points] Bob, the salesman at the shoe store, has a different model for foot growth. He gives Carla the formula B(t) = 5+6e t/8 for the length of Taser s feet t months since Carla took her first measurement. According to Bob s model, when will Taser s feet be 8 inches long? Give your answer in exact form with no decimals. Solution: We need to solve B(t) = 8 for t. 5+6e t/8 = 8 ln 8ln 8 5 6 8 5 8 = 5+6e t/8 = e t/8 = t/8 6 8 5 = t 6 Fall, 201 Math 115 Exam 1 Problem 1 (foot size) Solution