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Math Help - Comparing Quadratics

  1. #1
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    Comparing Quadratics

    Please help my solve this question: Two rockets are shot into the air from the same location. The paths of the rockets are given by
    #1: y = -0.025x^2 + 3x - 52.5,
    and #2: y = -0.04x^2 + 0.1x + 40.
    Which rocket goes higher, and by how much?

    I'm uncertain as to how I should solve this. Assistance would be much appreciated. Thank you.
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  2. #2
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    Hello, currypuff!

    Two rockets are shot into the air from the same location.

    The paths of the rockets are given by: . \begin{array}{ccc} y_1 &=& -0.025x^2 + 3x - 52.5 \\ y_2 &=& -0.04x^2 + 0.1x + 40\end{array}

    Which rocket goes higher, and by how much?

    Both graphs are down-opening parabolas; their maximums occur at their vertex.

    . . Vertex formula: . x \:=\:\frac{\text{-}b}{2a}


    For y_1\!:\;a = -0.025,\;b = 3\quad\Rightarrow\quad x \:=\:\frac{-3}{2(-0.025)} \:=\:60

    Hence: . y_1 \;=\;-0.025(60^2) + 3(60) - 52.5 \;=\;\boxed{37.5}


    For y_2\!:\;a = 0.04,\;b = 0.1\quad\Rightarrow\quad x \:=\:\frac{-0.1}{2(0.04)} \:=\:1.25

    Hence: . y_2\;=\;-0.04(1.25^2) + 0.1(1.25) + 40 \;=\;\boxed{40.0625}


    The second rocket goes 2.5625 units higher.

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