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Math Help - Optimization!

  1. #1
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    Optimization!

    When you cough, your windpipe contracts. The speed, v, with which air comes out depends on the radius, r, of your windpipe. If R is the normal (rest) radius of your windpipe, then for r ≤ R, the speed is given by the following formula, where a is a positive constant.

    v = a(R - r)r^2

    (a) Find v '(r).

    v '(r) =

    (b) What value of r maximizes the speed?

    r =


    so i believe to find the value of that r that maximizes speed, you would have to set the derivative equal to 0 and solve. However im having trouble finding the derivative, any help? thanks...
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  2. #2
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    Quote Originally Posted by mathlete View Post
    When you cough, your windpipe contracts. The speed, v, with which air comes out depends on the radius, r, of your windpipe. If R is the normal (rest) radius of your windpipe, then for r ≤ R, the speed is given by the following formula, where a is a positive constant.

    v = a(R - r)r2

    (a) Find v '(r).

    v '(r) =

    (b) What value of r maximizes the speed?

    r =


    so i believe to find the value of that r that maximizes speed, you would have to set the derivative equal to 0 and solve. However im having trouble finding the derivative, any help? thanks...
    Expand the RHS of the equation:

    v(r) = aRr^2-ar^3. ........... Calculate the first derivative:

    v'(r) = 2aRr - 3ar^2........... and set v'(r) = 0. Solve for r.

    I've got: r = 0 (this case is called strangulation) or r = \frac23R
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