Exponential Decay: Connecting Your Knowledge

Unit 3: Growing, Growing, Growing//Investigation 4//Connections Name _______________________________________ Class ______________ Date _______________...
Author: Barbra Stone
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Unit 3: Growing, Growing, Growing//Investigation 4//Connections Name _______________________________________ Class ______________ Date ______________________________

Exponential Decay: Connecting Your Knowledge I can recognize and express exponential patterns in equations, tables, and graphs.

Math ______ / 30 points Total ______ / 40 Points Reflection _______ / 10 points For this packet, your goal is to earn 40 points. 30 points will be from the math problems. 10 points will be from the reflection section. Look at the point values of the following questions and answer the questions that you feel most comfortable to answer.

Tell whether the equation represents exponential decay or exponential growth: 2 points

1. y = 0.9(3.3)x

2 points

2.

y = 25(0.3)x

2 points

3. y = 1.0(1.5)x

2 points

4.

y = 0.75(0.5)x

2 points

5. Which decay factor represents faster decay: 0.8 or 0.9? How do you know?

2 points

6. Which decay factor represents faster decay: 0.8 or 0.9? How do you know?

What is the decay factor for each relationship? 2 points

7.

2 points

8.



2 points

9.

What is the exponential equation that fits each of the tables above?

2 points

10. Equation for Question 7: ___________________

2 points

11. Equation for Question 8: ___________________

2 points

12. Equation for Question 9: ___________________

For questions 13 - 14 use Lara’s conjecture below. Explain how you found your answer.

5 points

13. The exponential decay graph has a y-intercept of 90, and it passes through the point (2, 10). When x = 1, what is y?

5 points

14. The exponential decay graph has a y-intercept of 40, and it passes through the point (2, 10). When x = 4, what is y?

5 points

15. Natasha and Michaela are trying to find growth factors for exponential functions. They claim that if the independent variable is increasing by 1, then you divide the two corresponding y values to find the growth factor. For example if (x1, y1) and (x2, y2) are two consecutive points, then the growth factor is y2 divided by y1. a. Is their reasoning correct? Explain why or why not.

b. Would this method work to find the growth pattern for a linear function? Explain why or why not.

5 points

16. Hot coffee is poured into a cup and allowed to cool. The difference between coffee temperature and room temperature is recorded every minute for 10 minutes.

a. What is the equation for the relationship shown in the table?

b. About how long will it take the coffee to cool to room temperature (21 degrees Celsius)? Show your work to prove your answer.

10 points

17. Karen shops at Aquino’s Groceries. Her bill came to $50 before tax. She used two of the coupons shown below.

Karen was expecting to save 10% (because she added the two coupons together), which is $5. The cashier rang up the two coupons. Karen was surprised when the total price rang up as $45.13 instead of $45 (before tax). She was not sure why the amount was higher than she anticipated. a. Why didn’t the coupons take off 10% in the way Karen expected?

b. Write an equation to represent the total amount (t) Karen would spend based on the number of coupons (n) she would use.

c. Karen had originally thought that if she used 10 coupons on her next bill of $50, she would save a total of 50%. How much would Karen ACTUALLY spend?

10 points

18. A cricket is on the 0 point of a number line, hopping toward 1. She covers half the distance from her current location to 1 with each hop. So, she will be at ½ after one hop, ¾ after two hops (because she is adding ¼ to the ½), and so on.

a. Make a table showing the cricket’s location for the first 10 hops (remember that each hop is ADDING ON to the previous hop to reach cricket’s actual location).

b. What equation can be used to represent where cricket will be after h hops?

c. Will the cricket ever get to 1? Explain.

10 points

19. Freshly cut lumber, known as green lumber, contains water. If green lumber is used to build a house, it may crack, shrink, and warp as it dries. To avoid these problems, lumber is dried in a kiln that circulates air to remove moisture from the wood. Suppose that, in 1 week, a kiln removes 1/3 of the moisture from a stack of lumber. Therefore, 2/3 of the moisture remains in the lumber. a. What fraction of the moisture remains in the lumber after 5 weeks in a kiln?

b. What fraction of the moisture has been removed from the lumber after 5 weeks in a kiln?

c. Write an equation for the fraction of moisture m remaining in the lumber after w weeks.

d. Write an equation for the fraction of moisture m has been removed from the lumber after w weeks.

e. Graph your equations from parts (c) and (d) on the same set of axes.

f.

How are your graphs related?

Exponential Decay: Reflection In this investigation, you explored situations in which a quantity decayed by a constant percent rate per unit interval and graphed them. You constructed exponential decay functions given a graph, a description, or a table. Answer the following questions in complete sentences. (This reflection is worth 10 of your total of 40 points.) 1.

How can you look at an equation and determine if it shows exponential growth or decay? Include an example of each.

2.

How is exponential decay shown in a table? Include an example.

3.

How are exponential decay functions and decreasing linear functions similar? How are they different?

4.

Create a real-life math problem that involves using exponential decay:

5.

Think about what you know about the relationship between a growth factor and a growth rate. Given that relationship, what do you think a decay rate is? What do you think the relationship would be between a decay factor and a decay rate?

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