ANSWERS QUESTIONS 9 AND 10 ON THE LAST PAGE. (THEY ARE STARRED)

EXTENDED TECHNOLOGY APPLICATION The Business of Motion Picture Revenue People access movies on different platforms, including tablets and smart phones, as well as disc formats such as DVD and Blu-ray. The timing of release dates for films on these secondary platforms is a major concern for movie executives. There has been increasing pressure to narrow the gap between the theatrical release of a movie and the release to secondary platforms. Movie executives want to reduce marketing expenses, and a studio realizes greater profit from selling a disc than from providing the Number of movie in streaming format. Theater Days Between owners, on the other hand, want to protect Movie Release Dates Release Dates their revenue and fear that with a shorter time between theater release and disc Hidden Figures Theater: January 6, 2017 95 release, more people will skip the theater Disc: April 11, 2017 presentation and wait. Timing of the disc Logan Theater: March 3, 2017 81 release becomes important to maximize Disc: May 23, 2017 the potential for profit. Gifted Theater: April 12, 2017 104 The table at the right gives t, Disc: July 25, 2017 the number of days between theater T2: Trainspotting Theater: March 31, 2017 88 release and disc release, for 10 movies. Note that the average time lag is about Disc: June 27, 2017 14 weeks, or a little over 3 months. The Lost City of Z Theater: April 21, 2017 81 Disc: July 11, 2017 Wonder Woman Theater: June 2, 2017 110 Disc: September 19, 2017 Star Wars: The Last Jedi Theater: December 15, 2017 103 Disc: March 27, 2018 Blade Runner 2049 Theater: October 6, 2017 103 Disc: January 16, 2018 The Post Theater: January 12, 2018 96 Disc: April 17, 2018 Hostiles Theater: January 26, 2018 88 Disc: April 24, 2018 Sources: imdb.com; boxofficemojo.com; dvdreleasedates.com.)260 CHAPTER 2 @& Exponential and Logarithmic Functions Let's examine data for Gifted. The table below presents weekly estimates of gross revenue, G, for that movie. Total revenue, R, is approximated by adding each week's gross revenue to the previ- ous week's total revenue. Revenue for Gifted Week in Release, t [ Weekly Revenue, G Total Revenue, R (week 1 began (estimates in (cumulative box-office 4/7/17) millions) revenue, in millions) 1, April 7 $1.37 $ 1.37 2, April 14 $4.84 $ 6.21 3, April 21 $6.32 $12.53 4, April 28 $4.66 517.19 5, May 3 $2.88 $20.07 6, May 12 $2.07 $22.14 7, May 19 $1.11 $23.25 8, May 26 $0.65 $23.90 9, June 2. ? ? 10, June 9 ? ? 11, June 16 ? 7 12, June 23 | ? ? 13, June 30 | ? ? (Sources: imdb.com; boxofficemojo.com; dvdreleasedates.com.) Exercises 1. A movie executive wants to fit a function to the data for 5. Repeat Exercise 4, ignoring the first two data points (that Gifted. Make a scatterplot of the data points (t, G); then use REGRESSION to fit linear, quadratic, cubic, and expo- nential functions to the data, and graph each equation with the scatterplot. Which function fits best? Why? 2. Use the exponential function to predict gross revenue, G, for weeks 9 through 13. 3. Use your predictions from Exercise 2 to compute total revenue, R, for weeks 9 through 13. 4. Use RecRression to fit a logistic function of the form R(t) = /(1 + ae) to the data points (t, R) for weeks 1-8. Based on these results, what seems to be a limiting value for the total revenue from Gifted? Does the value seem reasonable? . The disc release of Gifted occurred 15 weeks after its is, consider only weeks 3 through 8, and the total revenue, R, during those weeks). Does this function seem to be a better predictor of future earnings for Gifted? Why or why not? theatrical release. Based on the given data, do you think that this release date was appropriate? What factors might influence the release date of a movie onto disc formar? Extended Technology Application 261 Now consider the revenue data for Hidden Figures, shown Exercises in the following table. This movie was released in selected theaters on December 25, 2016 and released nationwide 7. Make a scatterplot of the data points (t, G) for weeks 1 (United States and Canada) on January 6, 2017. through 8, and fit linear, quadratic, cubic, and exponential functions to the data. Which function, if any, best esti- mates the weekly gross revenue for weeks 9 through 14? Revenue for Hidden Figures 8. Use the exponential model to predict the gross revenue, Week in Weekly Total Revenue, R G, for Hidden Figures for weeks 9 through 14. Release, Revenue, G cumulative box- 9. Use REGRESSION to fit a logistic function of the form t (week 1 (estimates in office revenue, R(t) = c/(1 + ae- of) to the data points (t, R) for began 1/6/17) millions) in millions) Hidden Figures for weeks 1 through 8. Based on these 1, Jan 6 $31.43 $ 31.43 results, what is the limiting value of the total revenue? Does this figure seem accurate, too high, or too low? 2, Jan 13 $33.61 $ 65.04 3, Jan 20 $22.03 $ 87.07 10. Repeat Exercise 9, but using only the data points for weeks 4, Jan 27 $19.28 $106.35 3 through 8. What is the limiting value of total revenue 5, Feb 3 $14.15 $120.50 from Hidden Figures? Does this seem to be a better esti- 6, Feb 10 $12.04 $132.54 mate? Why or why not? 7, Feb 17 $11.45 $143.99 11. A movie executive decides to release Hidden Figures to disc 8, Feb 24 $ 8.00 $151.99 when the gross theatrical revenues reach $160 million. 9, Mar 3 ? In what week should the executive plan to release Hidden 10, Mar 10 Figures in disc format? 11, Mar 17 IN IN 12. Why might it be wise to ignore the first two weeks of 12, Mar 24 data when finding a logistic function that models the 13, Mar 31 long-term behavior of a movie's total revenue? 14, Apr 7 (Sources: imdb.com; boxofficemojo.com; dvdreleasedates.com.)