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At the time that an article was first featured on the home page of a news website, there were 40 comments on the article. An exponential model estimates that at the end of each hour after the article was first featured on the home page, the number of comments on the article had increased by 190% of the number of comments on the article at the end of the previous hour. Which of the following equations best represents this model, where C is the estimated number of comments on the article t hours after the article was first featured on the home page ≤ 4 and t? = 40(1.19) t
Correct answer: D (parsed from the explanation — this source omits an answer key for this item)
Explanation
Choice D is correct. It’s given that an exponential model estimates that the number of comments on an article increased by a fixed percentage at the end of each hour. Therefore, the model can be represented by an exponential equation of = t the form C Ka, where C is the estimated number of comments on the article t hours after the article was first featured on the home page and K and a are constants. It’s also given that when the article was first featured on the home 40 page of the news website, there were comments on the article. This means = =40 40 = t that when t 0, C . Substituting 0 for t and for C in the equation C Ka 40= 0 40= yields Ka , or K. It’s also given that the number of comments on the 190% article at the end of an hour had increased by of the number of comments on the article at the end of the previous hour. Multiplying the percent increase by the number of comments on the article at the end of the previous hour yields the number of estimated additional comments the article has on its home page: 40 190 76 ` jb 100 l, or comments. Thus, the estimated number of comments for the following hour is the sum of the comments from the end of the previous hour and 40+76 116 the number of additional comments, which is , or . This means that = =116 116 40 when t 1, C . Substituting 1 for t, for C, and for K in the equation = t 116=40 1 116=40 C Ka yields a , or a. Dividing both sides of this equation by 40 2.9= 40 2.9 = t yields a. Substituting for K and for a in the equation C Ka =40 2.9 t yields C ^ h. Thus, the equation that best represents this model is =40 2.9 t C ^ h. Choice A is incorrect. This model represents a situation where the number of 19% comments at the end of each hour increased by of the number of comments 190% at the end of the previous hour, rather than . Choice B is incorrect. This model represents a situation where the number of comments at the end of each 90% hour increased by of the number of comments at the end of the previous 190% hour, rather than . Choice C is incorrect. This model represents a situation 19 where the number of comments at the end of each hour was times the number 190% of comments at the end of the previous hour, rather than increasing by of the number of comments at the end of the previous hour.
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