A cable company with a reputation for poor customer service is losing subscribers at a rate of approximately 3% per year. The company had 2 million subscribers at the start of 2014. Assume that the company continues to lose subscribers at the same rate, and that there are no new subscribers. Which of the following functions, S, models the number of subscribers (in millions) remaining t years after the start of 2014 ? Choose 1 answer: (A) S(t)=2(1.03)^(t) (B) S(t)=2(0.97)^(t) (c) S(t)=2(0.70)^(t) (D) S(t)=2(0.97)t

Question

A cable company with a reputation for poor customer service is losing subscribers at a rate of approximately  3% per year. The company had 2 million subscribers at the start of 2014. Assume that the company continues to lose subscribers at the same rate, and that there are no new subscribers. Which of the following functions,  S, models the number of subscribers (in millions) remaining  t years after the start of 2014 ? Choose 1 answer: (A)  S(t)=2(1.03)^(t) (B)  S(t)=2(0.97)^(t) (c)  S(t)=2(0.70)^(t) (D)  S(t)=2(0.97)t

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Find Decay Function: We need to find a function that models the decay of subscribers at a rate of 3% per year. The initial number of subscribers is 2 million. A decay rate of 3% means that each year, the company retains 97% of its subscribers from the previous year.Decay Formula: To model a percentage decrease, we use the formula S(t) = S_0 * (1 - r)^t, where S_0 is the initial amount, r is the rate of decrease (expressed as a decimal), and t is the time in years.Calculate Remaining Percentage: Since the company is losing 3% of subscribers each year, the rate r is 0.03. Therefore, the remaining percentage each year is 1 - r = 1 - 0.03 = 0.97.Substitute Values: Substituting the values into the decay formula, we get S(t) = 2 * (0.97)^t. This function represents the number of subscribers (in millions) remaining t years after the start of 2014.Compare with Options: Now we compare the function we found with the options given: (A) S(t)=2(1.03)^(t) - This represents an increase, not a decrease. (B) S(t)=2(0.97)^(t) - This matches our function. (C) S(t)=2(0.70)^(t) - This represents a much larger decrease than 3%. (D) S(t)=2(0.97)t - This is a linear function, not an exponential decay.Correct Function: The correct function that models the situation is therefore option (B) S(t)=2(0.97)^(t).

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