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T(n)=3.5+7*0.9^(n) The administrators of a factory modeled the cumulative average construction time, T(n), in minutes per engine component, as a function of the total number of components their employees had previously constructed, n, using the function shown. After the employees had constructed a very large number of engine components, what was the cumulative average time to construct a single component according to the model? Choose 1 answer: (A) 3.5 minutes per engine component (B) 7 minutes per engine component (C) 9.8 minutes per engine component (D) 10.5 minutes per engine component

T(n)=3.5+7⋅0.9n T(n)=3.5+7 \cdot 0.9^{n} \newlineThe administrators of a factory modeled the cumulative average construction time, T(n) T(n) , in minutes per engine component, as a function of the total number of components their employees had previously constructed, n n , using the function shown. After the employees had constructed a very large number of engine components, what was the cumulative average time to construct a single component according to the model?\newlineChoose 11 answer:\newline(A) 33.55 minutes per engine component\newline(B) 77 minutes per engine component\newline(C) 99.88 minutes per engine component\newline(D) 1010.55 minutes per engine component

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Full solution

Q. T(n)=3.5+7⋅0.9n T(n)=3.5+7 \cdot 0.9^{n} \newlineThe administrators of a factory modeled the cumulative average construction time, T(n) T(n) , in minutes per engine component, as a function of the total number of components their employees had previously constructed, n n , using the function shown. After the employees had constructed a very large number of engine components, what was the cumulative average time to construct a single component according to the model?\newlineChoose 11 answer:\newline(A) 33.55 minutes per engine component\newline(B) 77 minutes per engine component\newline(C) 99.88 minutes per engine component\newline(D) 1010.55 minutes per engine component
  1. Approaching 00: As nn approaches a very large number, the term 0.9n0.9^n approaches 00 because 0.90.9 is less than 11 and any number less than 11 raised to a large power gets closer and closer to 00.
  2. Function Simplification: So, the function T(n)=3.5+7×0.9nT(n) = 3.5 + 7 \times 0.9^n will approach 3.5+7×03.5 + 7 \times 0, because 0.9n0.9^n is almost 00.
  3. Calculating Limit: Now, calculate the limit of T(n)T(n) as nn approaches infinity: T(n)=3.5+7×0=3.5T(n) = 3.5 + 7 \times 0 = 3.5.
  4. Final Result: Therefore, the cumulative average time to construct a single engine component after constructing a very large number of components is 3.53.5 minutes per engine component.

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