An inferior product with a large advertising budget sells well when it is introduced, but sales fall as people discontinue use of the produc

Question

An inferior product with a large advertising budget sells well when it is introduced, but sales fall as people discontinue use of the product. Suppose that the weekly sales S are given by S = 500t /(t + 5)2 , t ≥ 0 where S is in millions of dollars and t is in weeks. After how many weeks will sales be maximized?

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Kinsley 4 months 2021-10-08T16:18:44+00:00 1 Answer 0 views 0

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    2021-10-08T16:20:25+00:00

    Answer:

    Sales will be maximized after 5 weeks .

    Step-by-step explanation:

    We are given that the weekly sales S are given by

    S=\frac{500t}{(t+5)^2}

    S'(t)=\frac{(t+5)^2\frac{d}{dt}(500t)-500t\frac{d}{dt}((t+5)^2)}{(t+5)^4}

    S'(t)=\frac{500(t+5)^2-500t(2(t+5))}{(t+5)^4}

    S'(t)=\frac{-500(t-5)}{(t+5)^3}

    Substitute the S'(t)=0

    So, S'(t)=\frac{-500(t-5)}{(t+5)^3}=0

    -500(t-5)=0

    t=5

    S'(t)=\frac{-500(t-5)}{(t+5)^3}\\S''(t)=\frac{(t+5)^3\frac{d}{dt}(-500(t-5))-(-500(t-5))\frac{d}{dt}((t+5)^3)}{(t+5)^6}\\S''(t)=\frac{(t+5)^3(-500)-(-500(t-5))(3(t+5)^2)}{(t+5)^6}

    Substitute t = 5 in S”(t)

    S''(t)=\frac{(5+5)^3(-500)-(-500(5-5))(3(5+5)^2)}{(5+5)^6}\\S''(t)=-0.5

    Since S”(t) at t =5 is less than 0

    So, maximum

    So, Sales will be maximized at t = 5

    Hence Sales will be maximized after 5 weeks .

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