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Difficulty: Hard

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 $190 percent sign$ 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 and $t less than or equals 4$ ?

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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 the form $C = K a^{t}$ , 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 page of the news website, there were $40$ comments on the article. This means that when $t equals 0$ , $upper C equals 40$ . Substituting $0$ for $t$ and $40$ for $C$ in the equation $C = K a^{t}$ yields $40 equals upper K a Superscript 0$ , or $40 equals upper K$ . It's also given that the number of comments on the article at the end of an hour had increased by $190 percent sign$ 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: $left parenthesis 40 right parenthesis left parenthesis StartFraction 190 Over 100 EndFraction right parenthesis$ , or $76$ 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 the number of additional comments, which is $40 plus 76$ , or $116$ . This means that when $t equals 1$ , $upper C equals 116$ . Substituting $1$ for $t$ , $116$ for $C$ , and $40$ for $K$ in the equation $C = K a^{t}$ yields $116 equals 40 a Superscript 1$ , or $116 equals 40 a$ . Dividing both sides of this equation by $40$ yields $2.9 equals a$ . Substituting $40$ for $K$ and $2.9$ for $a$ in the equation $C = K a^{t}$ yields $upper C equals 40 left parenthesis 2.9 right parenthesis Superscript t$ . Thus, the equation that best represents this model is $upper C equals 40 left parenthesis 2.9 right parenthesis Superscript t$ .

Choice A is incorrect. This model represents a situation where the number of comments at the end of each hour increased by $19 percent sign$ of the number of comments at the end of the previous hour, rather than $190 percent sign$ .

Choice B is incorrect. This model represents a situation where the number of comments at the end of each hour increased by $90 percent sign$ of the number of comments at the end of the previous hour, rather than $190 percent sign$ .

Choice C is incorrect. This model represents a situation where the number of comments at the end of each hour was $19$ times the number of comments at the end of the previous hour, rather than increasing by $190 percent sign$ of the number of comments at the end of the previous hour.