7.6b Exponential Growth Investigation: Situations That Don’t Start with One…
Algebra
Name ___________________________ per ___
Activity 1: Bacteria
1. Not all situations that involve exponential growth begin with only one bacteria or one principal knowing the information. Instead, there could be several bacteria that enter a cut before starting to reproduce.
Or the Principal and vice-principal could both know the important information before they both start to make calls to teachers. a. Recreate the table for the situation where one bacterium lands in a cut on your arm and begins to split into two new bacteria every 20 minutes.
Stage
Number of
Bacteria
0
1
1 2 3 4 5 6 7 8 9 10 b. What was the algebraic (y= or f(x)=) equation that models this situation? Use y for the total number of bacteria and x for the stage number. c. What was the ANS/LAST equation that modeled this situation? Include a start value. d. Imagine that instead of one bacterium landing in your cut, six bacteria land in your cut and begin to reproduce the same way, each bacterium splits into two new bacteria every 20 minutes. Fill in the table that models the situation through the first 10 stages.
Stage 0 1 2 3 4 5 6 7 8 9 10
Number of
Bacteria
6 e. Compare the two tables. How does stage #1 of the 1 st table (one bacterium) compare with stage #1 of the
2 nd table (six bacteria)? What mathematical relationship relates these two stages? f. Compare the two tables. How does stage #3 of the 1 st table (one bacterium) compare with stage #3 of the
2 nd table (six bacteria)? What mathematical relationship relates these two stages? g. Compare the two tables. How does stage #8 of the 1 st table (one bacterium) compare with stage #8 of the 2 table (six bacteria)? What mathematical relationship relates these two stages? nd h. Adjust your algebraic equations in part 1b to represent the situation in part d, where you started with 6 bacteria in your cut. Use y for the total number of bacteria and x for the stage number. i. What is the recursive (Ans/Last) equation that modeled the situation in part d (starting with 6 bacteria)? Include a start value.
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2. Instead of starting with 6 bacteria in your cut, assume that 15 bacteria are present from the beginning. a. What would be the new algebraic (y=) equation that would model this situation? Again, use y for the total bacteria and use x for the stage number. b. What would be the new recursive (ans/last) equation that would model this situation? Again, include a start value.
Activity 2: Phone Calls
3. Going back to the principal making three calls and each teacher making three calls, and so on. Assume that instead of just the principal knowing a total of 4 administrators know the important information and each administrator calls three teachers. Each of those teachers calls three teachers, and so on. a. What would be the new algebraic (y=) equation that would model this situation? Again, use y for the total number of teachers and use x for the stage number. b. What would be the new recursive (ans/las)t equation that would model this situation? Again, include a start value.
Activity 3: Working Backwards
4. Sometimes instead of writing the equation, you are given the equation and asked to interpret the situation. Assume that in the following equations they relate the number of total bacteria to the number of times the bacteria has split. Again, x is the stage number and y is the number of bacteria. a. If this equation was written, y = 30(2) x , how many bacteria did the cut start with? And what is the growth factor? b. If this equation was written, y = 64(4) x , how many bacteria did the cut start with? And what is the growth factor? (Note: this is not possible in the real world on earth (that we know of), it is just some hypothetical freaky situation.)
5. Discuss the following situations. Assume that in the following equations they relate the number of teachers to the stage number. Be sure to mention the number of principals/administrators who know from the start. a. If this equation was written, y = 3(2) x , how many principals/administrators were first present and how many phone calls does each principal/administrator make? What is the growth factor? b. If this equation was written, y = 2(5) x , how many principals/administrators were first present and how many phone calls does each principal/administrator make? What is the growth factor?
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Activity 4: Interpreting Equations and Finding Solutions
6. Finally, sometimes equations are written to find an answer to a specific situation. The following equations deal with the situation of 10 bacteria landing in a cut and doubling every 20 minutes. Your job: (use complete sentences.)
1) Tell what question the equation is asking AND
2) Then find the answer using your calculator. a. y = 10(2) 6 b. y = 10(2) 21 c. 40960 = 10(2) x d. 1,310,720 = 10(2) x
Activity 4: Interpreting Equations and Finding Solutions
6. Finally, sometimes equations are written to find an answer to a specific situation. The following equations deal with the situation of 10 bacteria landing in a cut and doubling every 20 minutes. Your job: (use complete sentences.) a. y = 10(2) 6
1) Tell what question the equation is asking AND
2) Then find the answer using your calculator. b. y = 10(2) 21 c. 40960 = 10(2) x d. 1,310,720 = 10(2) x
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e. 650000 < 10(2) x f. 1,000,000,000 > 10(2) x
7. Imagine this situation: The school has one principal and two administrators. They, collectively, decide to each to call 5 teachers and have each teacher call 5 other teachers, and so on. a. What would be the algebraic (y=) equation that would model this situation? b. Using the words LAST and ANSWER, write an equation to compute the number of newly informed teachers from one stage to the next stage? Include a start value. e. 650000 < 10(2) x f. 1,000,000,000 > 10(2) x
7. Imagine this situation: The school has one principal and two administrators. They, collectively, decide to each to call 5 teachers and have each teacher call 5 other teachers, and so on. a. What would be the algebraic (y=) equation that would model this situation? b. Using the words LAST and ANSWER, write an equation to compute the number of newly informed teachers from one stage to the next stage? Include a start value.
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