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Math Help - Advanced functions finding value of t that gives maximum

  1. #1
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    Advanced functions finding value of t that gives maximum

    I've been trying to find what time, t gives the maximum altitude for this question. I've found the answer using derivative way however i'm taking a advanced functions course and the teacher won't allow the calculus way so is there another way to find the value of t that gives the maximum altitude from this equation:

    h(t) = -16t^2 + 90t + 10 000
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  2. #2
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    Quote Originally Posted by Devi09 View Post
    I've been trying to find what time, t gives the maximum altitude for this question. I've found the answer using derivative way however i'm taking a advanced functions course and the teacher won't allow the calculus way so is there another way to find the value of t that gives the maximum altitude from this equation:

    h(t) = -16t^2 + 90t + 10 000
    Complete the square

    \displaystyle -16t^2+90t+10000=-16(t^2-\frac{90}{16}t+ \quad)+10000=-16\left(t^2-\frac{90}{16}+\left[ \frac{90}{32}\right]^2 \right)+10,000+16\left[ \frac{90}{32}\right]^2

    \displaystyle =-16\left(t-\frac{90}{32} \right)^2+\frac{162025}{16}

    Now the parabola is in vertex from so you can just read the max off.
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  3. #3
    Super Member Quacky's Avatar
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    h(t) = -16t^2 + 90t + 10 000
    Alternatively, you could think of the graph of this function.

    It's a negative parabola. Parabolas are symmetrical along their maxima or minima.

    So the t co-ordinate you require will lie along the quadratic's line of symmetry. The way to find the line of symmetry is straightforward:

    Put the function equal to 0 and solve, to find the t-intercepts.

    0 = -16t^2 + 90t + 10 000
    8t^2 - 45t - 5000 = 0

    This gives t\approx 27.97 or t\approx-22.34

    Not very nice.

    The line of symmetry will lie exactly half-way between them:

    \frac{27.97-22.34}{2} = 2.815

    This is the same result (approximately) that you'd get using calculus - for an exact answer, solve the quadratic exactly, leaving your answer in surd form - the surds should cancel and you get \displaystyle\frac{90}{32}. Not very impressive, but heck - it works.
    Last edited by Quacky; February 25th 2011 at 02:47 PM.
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