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Consider the following problem: The total number of subscribers Zhang Wei has for his video page is changing at a rate of r(t)=21-2t subscribers per week (where t is the time in weeks). At time t=8 weeks, Zhang Wei has 120 subscribers. How many subscribers does Zhang Wei have by week 20 ? Which expression can we use to solve the problem? Choose 1 answer: (A) r(20)-r(8)+120 (B) int_(8)^(20)r(t)dt+120 (C) r(20) (D) int_(20)^(20)r(t)dt

Consider the following problem:\newlineThe total number of subscribers Zhang Wei has for his video page is changing at a rate of r(t)=21−2t r(t)=21-2 t subscribers per week (where t t is the time in weeks). At time t=8 t=8 weeks, Zhang Wei has 120120 subscribers. How many subscribers does Zhang Wei have by week 2020 ?\newlineWhich expression can we use to solve the problem?\newlineChoose 11 answer:\newline(A) r(20)−r(8)+120 r(20)-r(8)+120 \newline(B) ∫820r(t)dt+120 \int_{8}^{20} r(t) d t+120 \newline(C) r(20) r(20) \newline(D) ∫2020r(t)dt \int_{20}^{20} r(t) d t

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Full solution

Q. Consider the following problem:\newlineThe total number of subscribers Zhang Wei has for his video page is changing at a rate of r(t)=21−2t r(t)=21-2 t subscribers per week (where t t is the time in weeks). At time t=8 t=8 weeks, Zhang Wei has 120120 subscribers. How many subscribers does Zhang Wei have by week 2020 ?\newlineWhich expression can we use to solve the problem?\newlineChoose 11 answer:\newline(A) r(20)−r(8)+120 r(20)-r(8)+120 \newline(B) ∫820r(t)dt+120 \int_{8}^{20} r(t) d t+120 \newline(C) r(20) r(20) \newline(D) ∫2020r(t)dt \int_{20}^{20} r(t) d t
  1. Understand the problem: Understand the problem.\newlineWe are given a rate of change of subscribers r(t)=21−2tr(t) = 21 - 2t and the number of subscribers at t=8t = 8 weeks, which is 120120. We need to find the total number of subscribers at t=20t = 20 weeks.
  2. Determine approach: Determine the correct approach to solve the problem.\newlineTo find the total number of subscribers at t=20t = 20 weeks, we need to integrate the rate of change from t=8t = 8 to t=20t = 20 and add the initial number of subscribers at t=8t = 8 weeks.
  3. Identify expression: Identify the correct expression to use.\newlineThe correct expression to use is the integral of the rate of change from t=8t = 8 to t=20t = 20, plus the initial number of subscribers at t=8t = 8 weeks. This corresponds to choice (B) ∫820r(t) dt+120\int_{8}^{20} r(t) \, dt + 120.
  4. Calculate integral: Calculate the integral of the rate function from t=8t = 8 to t=20t = 20. We need to integrate r(t)=21−2tr(t) = 21 - 2t from t=8t = 8 to t=20t = 20. ∫820(21−2t) dt=[21t−t2]820\int_{8}^{20} (21 - 2t) \, dt = [21t - t^2]_{8}^{20} = (21(20)−202)−(21(8)−82)(21(20) - 20^2) - (21(8) - 8^2) = (420−400)−(168−64)(420 - 400) - (168 - 64) = 20−10420 - 104 = −84-84

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