Accumulation with a Quadratic Function

Transcription

1 Accumulation with a Quadratic Function Julián Guerrera, who owns a small farm near San Marcos, Texas, has recently installed a small tank on his property to provide flood control for the creek that flows near his home. Under normal circumstances no water flows into the tank; however, on a particularly rainy day, the creek rises and begins to flow into the tank. Mr. Guerrera collects data from his water gauge every hour beginning when the water first starts running into the tank at 8: a.m.; he records the rate of flow in cubic meters per hour every hour for the first four hours of the storm. His table is provided below. The weather report predicts that the flood will continue to increase in its intensity until 1: p.m. and then quickly subside. Mr. Guerrera knows that if his tank, which can hold up to 22, cubic meters of water, is no more than half full at the end of the first five hours, his home is safe from the flooding water. Eduardo, Mr. Guerrera s son, is taking Algebra II,so he volunteers to monitor the gauge and to evaluate the data. He, of course, needs your help. The gauge readings taken from 8: a.m. to noon are recorded in the table provided below. 8: a.m. 9: a.m. 1: a.m. 11: a.m. 12: noon m 3 /hr. 7 m 3 /hr. 13 m 3 /hr. 23 m 3 /hr. 37 m 3 /hr. 1. The first task is to create a graph of the data. a) Code the data so that 8: a.m. is at time and so that m 3 /hr is represented by. b) Sketch a scatterplot of the data c) Label the axes with words. d) Draw a smooth continuous curve to connect the data points for this time period. 179

2 e) What type of function does the data appear to represent? 2. Since the flood is predicted to subside after hours, you need to know the rate at which the water will be flowing at 1: p.m.; however, you cannot wait until then to start moving furniture. You decide to take a chance on the fact that a quadratic function will fit the data relatively well. a) Enter the data into L1 and L2 on your graphing calculator and adjust the window to display the data. b) Calculate a quadratic regression on the data, and graph the function to make sure that it gives an acceptable fit. c) What is the function? d) Using your function, calculate the rate at which the water is flowing into the tank at 1: p.m. e) Add this data point to your table and to your graph. 3. The information that has been gathered represents the rate of flow of the water in cubic meters per hour, but you need to calculate the total amount of water that has collected in the tank over the five hour period. You make a quick phone call to your teacher, and she suggests that what you should approximate the area under the curve. a) Consider the units of the x-axis (time-axis) and the units of the y-axis (rate-axis). If you calculate the area of a rectangle on this grid, what units will be on the area? b) Following your teacher s suggestion, draw vertical line segments, one at each hour, from the time-axis to the curve. Now draw five horizontal segments beginning on the left-side of the curve to form rectangles so that the height of the first rectangle is, the second is 7, the third is 13, etc. (Since this is a continually increasing function, the rectangles will all lie completely below the curve.) c) What is the area (include the units) of the first rectangle? d) What is the area (include the units) of the second rectangle? 18

3 e) What is the area (include the units) of the third rectangle? f) What is the area (include the units) of the fourth rectangle? g) What is the area (include the units) of the fifth rectangle? h) Total the areas of the five rectangles, and explain what this measurement represents. i) Is this measurement too small or too large? Explain your choice based on the graph that you drew. j) Explain why your answer must be multiplied by 1 in order to answer the question about how much water is in the tank. 4. a) Draw another sketch of the data; however, this time sketch in rectangles that use the right-hand side of the region for the height. These are called right-hand rectangles b) What is the area (include the units) of the first rectangle? c) What is the area (include the units) of the second rectangle? d) What is the area (include the units) of the third rectangle? e) What is the area (include the units) of the fourth rectangle? f) What is the area (include the units) of the fifth rectangle? g) Total the areas of the five rectangles, and explain what this measurement represents. h) Is this measurement too small or too large? Explain your choice based on the graph that you drew. 181

4 . According to your calculation the amount of water in the tank will be between and. If the amount of water is greater than, the Guerrera family needs to prepare the house for flooding so the situation looks bad. Your under estimation looks acceptable; however, the over estimation means that you need to help move furniture to the second floor of the house.. You and Eduardo decide to refine your estimate instead of moving furniture; you need more data points in order to get rectangles with smaller widths. a) You make a new table and add in values for every half-hour by calculating the rate predicted by your function at., 1., 2., 3., and b) You sketch a new graph and draw in 1 right-hand rectangles c) The new upper bound for the amount of water in the tank is. This is better, but it still predicts that you must move furniture. 182

5 7. To simplify your work, you decide to use the mathematical notation for this process which is last t-value f() t t. first t-value, (step= t) This says to sum the product of the height and the width of the rectangles beginning at t = first t-value and stopping at t = making a calculation every t units. t =, where n is the number of rectangles that you want to use. n To use this notation for the set of left-hand rectangles you write 4 t=, (step = 1) 2 (2t + 1) 1= (Notice that is the first value of t for the left-hand rectangles) To use this notation for the set of right-hand rectangles you write t= 1, (step = 1) 2 (2t + 1) 1 = (Notice that 1 is the first value of t for this set of right-hand rectangles) The set of 1 right-hand rectangles is written as t=., (step =.) 2 (2t 1). + = a) Explain why. is the first t-value for this set of right-hand rectangles. b) Compete the expanded version of this statement (2(.) + 1). + (2(1) + 1). + (2(1.) + 1)

6 8. You remember that there is a way to do this with your calculator, but you cannot remember the details. Again you call your teacher for assistance, and she reminds you that the calculator has a set of commands that will allow you to sum a sequence. She tells you to make sure that your equation is entered y1 of your calculator and then to go back to the home screen. On the calculator f() t t is entered as t= t,(step = t) sum(seq (y1 x, x, first x,, x) ) You decide to check that you understand the process by repeating your calculation of 1 right-hand rectangles. t = =., and the first t-value is.. 1 Remember that in your calculator the equation is written as a function of x instead of t. On a TI83 calculator: Press 2 nd Stat List Arrow over to Math Choose # Sum( Press 2 nd Stat List again Arrow over to OPS Choose # Seq( Your screen now shows sum(seq( Press VARS Arrow over to Y-VARS Choose #1 Function Select #1 Y1 Your screen now shows sum(seq(y1 Type., comma Complete the statement by typing x, comma,., comma,, comma, and., close parenthesis. Press enter to get the total area with 1 rectangles. Remember that you have coded your data so that you will need to multiply the final answer by 1 to get back to the original units. Check to be sure that your answers agree. 184

7 9. You decide to try using 2 rectangles. a) What is the value of t for 2 rectangles? b) What is the first t-value for 2 right-hand rectangles? c) What is the mathematical notation for 2 right-hand rectangles? d) Using your calculator, calculate the amount of water predicted by 2 right-hand rectangles? Do you still have to move furniture? 1. Continue the process, doubling the number of rectangles, until you can show that the amount of water in the tank at 1: p.m. will be less than 11, cubic meters. a) Your final predicted amount of water is. b) The final number of rectangles used is. c) The final width of each rectangle is. 11. Write a summary of the results of the activity that explains the thinking process used to get the final results. 18