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Math Help - Major Calculus Assignment Issue

  1. #1
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    Major Calculus Assignment Issue

    I know I put one of the questions up earlier, but it would really help me out if someone could give me some input on the rest of these.

    1. (a) Determine the area of the largest rectangle that can be inscribed in a right triangle with legs adjacent to the right angle of length 5 cm and 12 cm. The two sides of the rectangle lie along the legs.
    (b) Repeat part (a) with a right triangle that has sides 8 cm by 15 cm.
    (c) Hypothesize a conclusion for any right triangle.

    2. Determine the dimensions of the cone of maximum volume that can be inscribed in a sphere with radius 3?

    3. The position, s, of an object moving in a straight line from a fixed point at t seconds is given by
    s(t)=t^3 - 9t^2 - 21t - 11, where t>0 and s(t) is measured in metres.
    (a) Determine when the object is stopped and when it is moving forward.
    (b) Is the acceleration negative at any time?
    (c) When is the object speeding up and when is the object slowing down? Explain why.



    Any help would do wonders.
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  2. #2
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    Quote Originally Posted by philipsach1 View Post

    3. The position, s, of an object moving in a straight line from a fixed point at t seconds is given by
    s(t)=t^3 - 9t^2 - 21t - 11, where t>0 and s(t) is measured in metres.


    You have, for t>0
    s(t)=t^3-9t^2-21t-11
    Thus,
    s'(t)=v(t)=3t^2-18t-21
    Thus,
    s''(t)=v'(t)'=a(t)=6t-18

    (a) Determine when the object is stopped and when it is moving forward.
    The object is at rest when the speed is zero.
    Thus,
    3t^2-18t-21
    Divide by three,
    t^2-6t-7=0
    (t-7)(t+1)=0
    t=7,-1
    But time is measure positive here,
    t=7.

    (b) Is the acceleration negative at any time?
    We need that,
    6t-18<0
    Thus,
    6t<18
    Thus,
    t<3

    (c) When is the object speeding up and when is the object slowing down? Explain why.
    Look at (b) the object slowing down when acceleration is zero (decelleration) so it is slowing down until 3 seconds.
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  3. #3
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    Sorry, your Web browser is not Java compatible. This prototype will not be of much interest to you.
    I hope this helps you get a visual of the first problem...

    Move the red points and see when the area is the greatest














    Tell me if this helps
    Last edited by Quick; November 1st 2006 at 05:45 PM. Reason: adding labels
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  4. #4
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    Hello, Philip!

    2. Determine the dimensions of the cone of maximum volume
    that can be inscribed in a sphere with radius 3
    Code:
                    A
                  * * *
              *    /:\    *
            *     / : \     *
           *     /  :3 \     *
                /   :   \
          *    /    :    \    *
          *   /    O*     \   *
          *  /   *  :  * 3 \  *
            / *    y:     * \
         B *--------+--------* C
            *           x   *
              *           *
                  * * *

    The center of the sphere is O.
    Note that OA = OB = OC = 3.
    \Delta ABC is the side view of the cone.

    The volume of a cone is: . V \:=\:\frac{1}{3}\pi r^2h

    In the diagram, the radius is x and the height is y + 3
    So we have: . V \:=\:\frac{1}{3}\pi x^2(y+3) [1]

    The diagram also says: . x^2 + y^2 \:=\:3^2\quad\Rightarrow\quad x^2 \:=\:9 - y^2 [2]

    Substitute [2] into [1]: . V \;= \;\frac{1}{\pi}(9 - y^2)(y + 3)


    We have: . V \;=\;\frac{\pi}{3}\left(-y^3 - 3y^2 + 9y + 27\right)

    Then: . V' \;=\;\frac{\pi}{3}\left(-3y^2 - 6y + 9\right) \:=\:0

    We have the quadratic: . y^2 + 2y = 3 \:=\:0

    . . which factors: . (y - 1)(y + 3)\:=\:0

    . . and has roots: . y = 1,\,-3

    Since the measurements are positive, y = 1
    . . and from [2]: . x^2 \:=\:9-1^2\:=\:8\quad\Rightarrow\quad x \,= \,2\sqrt{2}


    Therefore, the cone of maximum volume has radius 2\sqrt{2} and height 4.

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  5. #5
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    Thanks!

    Thank you all so very much. All of these were of great help to me. :>
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  6. #6
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    Quote Originally Posted by philipsach1 View Post
    Thank you all so very much. All of these were of great help to me. :>
    I'm just curious (since I'm experimenting with the drawing program) did my diagram help at all? please be honest
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  7. #7
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    Re:

    It really was helpful. It helped me to visualize the problem. But since I had to look for dimensions, it would have been better if those had been on there, but it was still very helpful. Thank you again.
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