Your baby brother, Zach, is starting to talk and you start keeping a list of how many words he knows. When Zach was 12 months old, he had a vocabulary of 25 words and the list of words increases exponentially at a rate of 5% per week. Do you expect him to have a vocabulary of at least 300 words by his next birthday? Explain your answer. (Hint: Write an exponential function and evaluate that function for x = 52 since there are 52 weeks in a year.)

A. No. Zach will only have 285 words in his vocabulary.
Incorrect. A linear function y = 25 + 5x was used to model the growth. Using x = 52, Zach’s brother would know 285 words by his next birthday.

B. No. Zach will only have less than 30 words in his vocabulary.
Incorrect. The correct exponential equation y = 25〖(1+0.05)〗^x was used but 2 was substituted in for x, thinking Zach would be 2 years old. The growth rate is per week.

C. There is not enough information to make a determination.
Incorrect. There is enough information.

Yes. Zach’s vocabulary will exceed 300 words by the time he turns two.
Correct! The following functions were entered into the calculator: y₁ = 25(1 + .05)^x and y₂ = 300.

The number of telecommuters, or people who spend at least part of their work day at home and who use computers or some other telecommunication equipment, has grown exponentially, since 1990, as indicated by the table below.

Find an exponential model that fits the data and use your model to predict the number of telecommuters in the year 2020. Round your answer to the nearest million people.

A. 99,000,000 telecommuters
Incorrect. The correct exponential function y = 4.36(1 + .169)^x was used; however, ‘20’ was substituted for x instead of 30. Remember x represents the number of years that have lapsed since 1990.

B. 472,000,000 telecommuters
Correct! The exponential model is y = 4.36(1 + .169)^x. Let x = 30 since 30 years will lapse from 1990 to 2020. Put the equation into your graphing calculator. Use either the TABLE or the Home Screen to find the number of telecommuters after 30 years, 472 million.

C. 83,000,000 telecommuters
Incorrect. The quadratic model y = .219x² − .479x + 5.60 was used to model the data and ‘20’ was used for x instead of 30. The growth was exponential so the quadratic model is incorrect.

D. 188,000,000 telecommuters
Incorrect. The quadratic model y = .219x² − .479x + 5.60 was used to model the data, the growth was exponential so the quadratic model is incorrect.

A new laptop computer, originally valued at $1150, decreases at the rate of 12% per year. Suppose your friend comes to you 6 months after you purchase your laptop and wants to buy it used. He offers you $1,000 for it. Would that be a good deal? Why or why not?

A. $1,000 would not be a good deal because the laptop would still be valued more than $1,000.
Correct! The exponential function is y = 1150(1 −.12)x, where x is in years. Let x = .5 since the friend wants to buy it six months after it was purchased. Put the equation in your calculator and use the TABLE features to find the laptop’s value after six months. It would be $1078.80, which exceeds $1000.

B. $1,000 would be a good deal because the laptop’s value would only be worth about $530.
Incorrect. The correct exponential function was used; however, ‘6’ was substituted in for x. Remember, x is measured in years.

C. $1,000 would not be a great deal because the value of the laptop would be $1494.94.
Incorrect. The exponential equation is y = 1150 (1 − .12)^x.

D. There’s not enough information to make an informed decision.
Incorrect. There is sufficient information.

Estimates, in millions, of the number of U.S. households with digital televisions from 2003 to 2007 are shown in the table.

Find an exponential growth model that fits the given data. Then use your model to approximate the year you would expect the number of U.S. households that have digital televisions to exceed 200 million?

A. Between 2012 and 2013.
Incorrect. The growth model is y = 43.89(1.13)^x is used. Putting the function into the calculator the table shows 200 million occurs between 12 and 13, but, remember the starting value is for 2003. Therefore it is 12 – 13 years after 2003.

B. Close to the year 2026.
Incorrect. The linear growth model y = 43.04 + 6.95x is used instead of an exponential model.

C. There is insufficient information to answer this question.
Incorrect. There is sufficient information. Use the equation y = 43.89(1.13)^x.

D. Between 2015 and 2016.
Correct! The correct growth model is y = 43.89(1.13)^x.