Question

Unit 4M : Modeling and Analyzing Exponential Functions "Growing Exponentially -Performance Task #2 Be sure to show your work where possible-failure to show work will result in deduction in grade Part I: Meet Jan Jan’s lifelong dream has been to open her own business. After working, sacrificing, and saving, she finally has enough money to open up an ice cream business. The grand opening of her business is scheduled for the Friday of Labor Day weekend. She would like to have a soft opening for her business on the Tuesday before. The soft opening should give her a good idea of any supply or personnel issues and give her time to correct them before the big official opening. A soft opening means that the opening of the business is not officially announced; news of its opening is just spread by word of mouth (see, not all rumors are bad!). Jan needs a good idea of when she should begin the rumor in order for it to spread reasonably well before her soft opening. She has been told that about 10% of the people who know about an event will actually attend it. Based on this assumption, if she wants to have about 70 people visit her store on the Tuesday of the soft opening, she will need 700 people to know about it. 1. Jan plans to tell two people each day and will ask that person to tell two other people each day through the day of the opening, and so on. Assume that each new person who hears about the soft opening is also asked to tell two other people each day through the day of the opening and that each one starts the process of telling their friends on the day after he or she first hears. When should Jan begin telling others about the soft opening in order to have at least 700 people know about it by the day it occurs? 2. Let x represent the day number and let y be the number of people who know about the soft opening on day x. Consider the day before Jan told anyone to be Day 0, so that Jan is the only person who knows about the opening on Day 0. Day 1 is the first day that Jan told two people about the opening. a. Complete the following table. Day0 1 2 3 4 5 Number of people who know 1 3 13. Graph the points from the table in part 2a. a. Does it make sense to connect the dots on the graph? Why or why not? b. What does point (5, 32) represent in this situation? Describe the point in a form of a function notation. 2c. Describe the domain of the function. What does the domain mean in this situation? What are the restrictions of the domain that arise from the context? d. Describe the range of the function. What does the range mean in context of the situation? What are the restrictions of the range that arise from the context? e. What is the y-intercept in this problem? What does the y-intercept represent in this situation? 4. The scenario above can be modeled by a type of function known as exponential function; in particular, an exponential growth function. An exponential function has the form , , where a is a non–zero real number and b is a positive real number other than 1. An exponential growth function has a value of b that is greater than 1. Values “a” and “b” are the parameters of the function. Write explicit and recursive equations that describe the relationship between x (day) and y(number of people who know) for the situation of spreading the news about the soft opening of Jan’s ice cream store. What type of a sequence does the function represent? What do the parts of the equations represent in terms of the context?

Answer 1

1)

From the information given,

x represents the day number

y represents the number of people who know about the opening on day x

On the first day, x = 0, y = 1

On the second day, x = 1, y = 3

On the third day, x = 2, y = 9

The rate at which the number of people is increasing is exponential. The general form of an exponential equation is

y = ab^x

We would find a and b by substituting corresponding values of x and y into the equation. We have

For x = 0 and y = 3,

3 = ab^0

a = 3

For x = 1 and y = 3,

3 = ab^1

By substituting a = 3,

3 = 3b

b = 3/3 = 1

The equation is