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Math Help - the firework

  1. #1
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    the firework

    the equation for a firework is h=-5d2+20d+1

    h=height above the ground
    d=horizontal distance

    What would the max height of the firework be?
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by mcgill33 View Post
    the equation for a firework is h=-5d2+20d+1

    h=height above the ground
    d=horizontal distance

    What would the max height of the firework be?
    I am going to assume two things..firstly d2=dē and secondly, that you are in calc

    You are attempting to maximize h(d)=-5d^2+20d+1

    differentiating we get

    h'(d)=-10d+20

    So now we need to find where this equals zero

    h'(d)=0\Rightarrow{-10d+20=0}\Rightarrow{d=2}

    to verify this is a max we use the second derivative test

    and since h''(d)<0,\forall{x}\in\mathbb{R}
    we can see that this is in fact a max
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  3. #3
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    Judging from the questions you posted, this is probably what you want done:

    h=-5d^{2}+20d+1

    Complete the square to get your equation in the form of: h = a(d - b)^{2} + c

    where the sign of a determines which way the parabola points up (should be negative if you're looking for a max) and (b, c) is your vertex.

    The vertex should be the highest point of the parabola and that'll be your answer.
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  4. #4
    A riddle wrapped in an enigma
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    Max value can be determined by finding the x-coordinate of the vertex and then substituting that into the original function:

    y=ax^2+bx+c

    x=\frac{-b}{2a}

    f(x)=-5x^2+20x+1

    x=\frac{-20}{-10}=2

    f(2)=-5(2)^2+20(2)+1 = 21 which is the max height.
    Last edited by masters; June 4th 2008 at 11:40 AM.
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  5. #5
    Lord of certain Rings
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    Quote Originally Posted by mcgill33 View Post
    the equation for a firework is h=-5d2+20d+1

    h=height above the ground
    d=horizontal distance

    What would the max height of the firework be?
    Or By algebra:

    h=-5d^2+20d+1 = -5(d^2 - 4d) + 1 = -5(d^2 - 4d + 4) + 21 = 21 - 5(d-2)^2

    Now since (d-2)^2 \geq 0 always, h = 21 - 5(d-2)^2 \leq 21

    Equality holds when d=2 and the maximum height is h = 21
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