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Math Help - A couple calc questions [optimization, max/min/inflection]

  1. #1
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    A couple calc questions [optimization, max/min/inflection]

    The depth of water at my favorite surfing spot varies from 4 to 1 feet, depending on time. Last sunday, high tide occurred at 5:00 a.m. and the next high tide occurred at 6:30 p.m. Obtain a cosine model describing the depth of the water as a function of time t in hours since 5 a.m. Sunday morning. Find the fastest rate at which the tide was rising on Sunday. Waht what time (after 5 a.m Sunday) did that first occur?
    -I managed to get the cosine model, but I'm not sure where to go from there. I have y=1.5cos(13.5t)+5.5 as the model.

    The function f(x) = x cos( ln x ) has infinitely many critical points. Find the smallest x-value in the interval [1,∞) for which f(x) a relative minimum point, a relative maximum point, and an inflection point.
    -I'm not sure where to go if the function has infinitely many critical points. I know you're supposed to derivate, set the function equal to zero (finding critical points), and then check to see where it has max/mins, but if it has infinitely many I'm not sure where to go with that.

    Find the dimensions of the right circular cylinder of greatest volume that can be inscribed in a sphere of radius a.
    -I drew the diagram but I'm having a hard time trying to figure out where to go from there.
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  2. #2
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    Dear jelloish,

    I can see a contradiction:
    1) varies from 4 to 1 feet
    2) y=1.5cos(13.5t)+5.5 --> from 4 to 7

    The function f(x) = x cos( ln x ) has indeed infinitely many critical points what is easy to find with the derivation. But it hasn't global min or max because of x multiplying.

    "I drew the diagram..." --> That is very well!
    Let be r the radius of cylinder. Than try to find out the high (h) of cylinder from the fact: "can be inscribed in a sphere of radius a". Oke, now express the volume of cylinder on depending r:
    V(r) = r*r*pi*h(r)
    This is the function. Where is its maximum?
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  3. #3
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    correction for your model ...

    y = 1.5\cos\left(\frac{4\pi}{3} \cdot t\right) + 2.5

    \frac{dy}{dt} is the rate of change of the tide level

    \frac{dy}{dt} = -2\pi \sin\left(\frac{4\pi}{3} \cdot t\right)

    this is the function you want to maximize ... it will have a maximum positive value when \sin\left(\frac{4\pi}{3} \cdot t\right) = -1.


    for the second problem, find the smallest critical value that is \geq 1

    f(x) = x \cdot \cos(\ln{x})

    f'(x) = -x \sin(\ln{x}) \frac{1}{x} + \cos(\ln{x})

    f'(x) = \cos(\ln{x}) - \sin(\ln{x})

     f''(x) = -\frac{1}{x} [\sin(\ln{x}) + \cos(\ln{x})]

    for critical values where f'(x) = 0 , think where \cos(\ln{x}) = \sin(\ln{x}) ...

    \ln{x} = \frac{\pi}{4} \, , \, \frac{5\pi}{4} \, , \, \frac{9\pi}{4} \, ...

    the first max is at x = e^{\frac{\pi}{4}} and the first min is at x = e^{\frac{5\pi}{4}}.
    you can use the 2nd derivative test to confirm.


    for the first inflection point ...

    \sin(\ln{x}) = -\cos(\ln{x})

    \ln{x} = \frac{3\pi}{4}

    x = e^{\frac{3\pi}{4}}
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    Correction! I must have copied the question down wrong, for the tide question, the spot varies from 4 to 17 feet. I forgot the 7 when copying down the problem and the 1 when I was solving the problem. I have the correct model now- y=6.5cos[13.5t]=10.5. Skeeter, how did you get 4pi/3 for the center part of the model?
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    Skeeter, I got to the part cos(lnx)=sin(lnx) by myself when solving the third problem but I wasn't sure how to solve that. How exactly did you get the e^pi/4, etc.?
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    Quote Originally Posted by Skalkaz View Post

    "I drew the diagram..." --> That is very well!
    Let be r the radius of cylinder. Than try to find out the high (h) of cylinder from the fact: "can be inscribed in a sphere of radius a". Oke, now express the volume of cylinder on depending r:
    V(r) = r*r*pi*h(r)
    This is the function. Where is its maximum?
    I'm not sure how to find the max because there aren't any numerical values. I know you find the derivative of the function, find critical points, and test them to see if they're max/mins, but if I can't find the actual "points" because there are no values, I'm not sure how to go about doing that.

    If V(r)=r^2*pi* h(r), how do I derivate h(r)?
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    sin(ln(x)) = cos(ln(x)) --> tan(ln(x)) = 1
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    If you make the profil as sketch, you will give a circle with radius a and rectangle with side 2*r and h(r). Isn't true?

    Try to find a connection between this 3 values and express h(r). Viva geometry!!!
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  9. #9
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    my mistake ... I saw 5:00 AM and thought the next high tide was at 6:30 AM, not PM as I see now.

    period = 13.5

    B = \frac{2\pi}{13.5} = \frac{4\pi}{27}

    so ... readjusting your amplitude and period

    y = 6.5\cos\left(\frac{4\pi}{27} \cdot t\right) + 10.5
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