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Berry is standing at the edge of a hill that slopes downward and to the right. He notices that for every 5 feet you go in a horizontal direction, the hill drops by 2 feet. Berry places a ball at the edge of the hill (the origin of the coordinate system), and kicks the ball at a speed of 20 m/s and at an angle of 30o to the horizontal. The equation describing the ball’s height as a function of the horizontal distance it travels is given by . How far down the hill does the ball land?

  • precalculus -

    the velocity has horizontal and vertical components
    Vx = 20(√3/2) = 17.32
    Vy = 20(1/2) = 10.00

    The horizontal and vertical positions of the ball are thus

    sx = 17.32t
    sy = 10.00t - 4.9t^2
    So, the x-y function of the height is
    y = 10.00(x/17.32) - 4.9(x/17.32)^2
    = -0.01633x^2 + 0.5774x

    The ground is descending with slope = -2/5 so the surface is given by

    y = -2/5 x

    So, where does the parabola intersect the line?

    -0.01633x^2 + 0.5774x = -0.4x
    x = 59.85
    y = -23.94

    so the distance down the slope is

    d = √(59.85^2 + 23.94^2) = 64.46 m

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