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Math Help - Couple of differentiation questions

  1. #1
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    Couple of differentiation questions (have added second problem)

    Hi, I'm a bit stuck on a couple of things.


    And the second one:
    The diagram shows part of the curve with equation y=f(x)
    It's basically an N shaped quadratic with a maximum at B. The curve cuts the x-axis at the points A and C.
    f(x) = 200 - (250/x) - x^2, x>0
    Find f'(x) I've done this, 250x^-2 -2x
    Use this answer to calculate the co-ordinates of B.
    I'm not really sure what to do. Usually these kind of things would factorise?

    First question: HAVE SORTED THIS NOW, THANKS!
    A cylindrical biscuit tin has a close-fitting lid which overlaps the tin by 1 cm. The radii of the tin and the lid are both x cm. The tin and the lid are made from a thin sheet of metal of area 80pi cm^2 and there is no wastage. The volume of the tin is V cm ^3.
    Part
    a) is show that V = pi(40x-x^2-x^3)
    which I've done.
    b) use differentiation to find the positive value for which V is stationary. I've got this to be 10/3, which I think is right.
    c) Prove that this gives a maximum value of V.
    I've done this ok, too.
    d) Find this maxium value of V. I'm probably being really stupid, but this is the bit I can't do. I've tried a couple of things but I'm not coming out with the right answer.
    e) The last bit is 'determine the percentage of the sheet metal used in the lid when V is a maximum'. Also not sure where to start with this. Can anyone give me any pointers?
    Last edited by Selene; March 11th 2009 at 01:52 PM.
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  2. #2
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    Quote Originally Posted by Selene View Post
    Hi, I'm a bit stuck on a couple of things.

    First question:
    A cylindrical biscuit tin has a close-fitting lid which overlaps the tin by 1 cm. The radii of the tin and the lid are both x cm. The tin and the lid are made from a thin sheet of metal of area 80pi cm^2 and there is no wastage. The volume of the tin is V cm ^3.
    Part
    a) is show that V = pi(40x-x^2-x^3)
    which I've done.
    b) use differentiation to find the positive value for which V is stationary. I've got this to be 10/3, which I think is right.
    c) Prove that this gives a maximum value of V.
    I've done this ok, too.
    d) Find this maxium value of V. I'm probably being really stupid, but this is the bit I can't do. I've tried a couple of things but I'm not coming out with the right answer.
    e) The last bit is 'determine the percentage of the sheet metal used in the lid when V is a maximum'. Also not sure where to start with this. Can anyone give me any pointers?
    For d) substitute the x-value you found gives the maximum into the equation V = \pi(40x - x^2 - x^3)
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  3. #3
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    Ah, yep. Done them both now! Got (2300pi)/27 and 22 2/9 % for e.
    Was having a bit of a moment!
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