1 TASK 3.3.4: EXPONENTIAL DECAY NO BEANS ABOUT IT A genie has paid you a visit and left a container of magic colored beans with instructions. You are to locate the magic bean for your group. You will be given boxes in which to place the beans. You will remove one third of the beans and place these in box 1. Then, you will remove one third of the beans in box 1 and place these beans in the second box. In the third box you will place one third of the beans from box 2. You will continue this process until have only one bean. This is your magic bean. This bean can be used to create more magic beans. When the genie left you with your instructions the genie said only whole beans can be removed from the box. 1. Suppose the container has 2700 beans. Complete the table. Remember you can only remove whole beans, so if you get a decimal answer, use only the whole number portion and do not round. Table I Box Process Number of Beans 1 2700*( 1 3 )1 900 2 2700*( 1 3 )2 300 3 2700*( 1 3 )3 100 4 2700*( 1 3 )4 33 5 2700*( 1 3 )5 11 6 2700*( 1 3 )6 3 7 2700*( 1 3 )7 1 n 2700*( 1 3 )n
2 2. Find a viewing window for the problem situation. Sketch your graph: Identify your window: Xmin: 1 Xmax: 8 Xscl: 1 Ymin: 0 Ymax: 900 Yscl: 1 3. Justify your window choices. Sample answer: The variable x stands for the number of boxes, so one to eight boxes appear reasonable. The variable y stands for the number of beans removed and placed in the next box, so 0 through 900 will show the number beans for boxes 1 through 7. 4. What if the genie gave you 1 billion magic beans? Write a function for how many beans will be in box n. b = 1,000,000,000*( 1 3 )n 5. Instead of one billion beans, assume the genie gave you an original container with 16,200 beans. Using the same rule, in which box would you have 200 beans? Box 4 16,200*( 1 3 ) 1 = 5400 Box 1 16,200*( 1 3 ) 2 = 1800 Box 2 16,200*( 1 3 ) 3 = 600 Box 3 16,200*( 1 3 ) 4 = 200 Box 4
3 6. a. Suppose you have a box with 5 beans and you know it is box number 10. How many beans were in the original container if in the decay process you lose 2 3 each time? 295,245. To find the answer, note that losing 2 3 means that the box will contain 1 3 of the original. Therefore you can reverse the process by multiplying each time by 3.. 5 Box 10 15 Box 9 45 Box 8 135 Box 7 405 Box 6 1215 Box 5 3645 Box 4 10,935 Box 3 32,805 Box 2 98,415 Box 1 295,245 Original Container b. Write a function rule to find the number of beans in the original container given the beans in the n th container. b = 5*3 n 7. In these activities you have been working with exponential growth and exponential decay. List some similarities between the two functions and list some major differences. Similarities: In both functions, the operation is multiplication, ratios are constant, both written the same way with a starting point, equations written the same, and have 1 as a common point. Differences: Differences are that in exponential growth the multiplier is greater than 1 and in exponential decay the multiplier is less than one. In exponential decay the function is approaching a limit of zero. In exponential growth the function is approaching positive infinity.
4 TASK 3.3.4: EXPONENTIAL DECAY NO BEANS ABOUT IT A genie has paid you a visit and left a container of magic colored beans with instructions. You are to locate the magic bean for your group. You will be given boxes to place the beans. You will remove one third of the beans and place these in box 1. Then, you will remove one third of the beans in box 1 and place these beans in the second box. In the third box you will place one third of the beans from box 2. You will continue this process until have only one bean. This is your magic bean. This bean can be used to create more magic beans. When the genie left you with your instructions the genie said only whole beans can be removed from the box. 1. Suppose the container has 2700 beans. Complete the table. Remember you can only remove whole beans, so if you get a decimal answer, use only the whole number portion and do not round. Table I Box Process Number of Beans 1 2 3 4 5 6 7 n
5 2. Find a viewing window for the problem situation. Sketch your graph: Identify your window: Xmin: Xmax: Xscl: Ymin: Ymax: Yscl: 3. Justify your window choices. 4. What if the genie gave you 1 billion magic beans? Write a function for how many beans will be in box n. 5. Instead of one billion beans, assume the genie gave you an original container with 16,200 beans. Using the same rule, in which box would you have 200 beans? 6. a. Suppose you have a box with 5 beans and you know it is box number 10. How many beans were in the original container if in the decay process you lose 2 3 each time? b. Write a function rule to find the number of beans in the original container given the beans in the n th container. 7. In these activities you have been working with exponential growth an exponential decay. List some similarities between the two functions and list some major differences.