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Highway engineers often use quadratic functions to model safe distance for cars. For example, f(x)= 1/2 x^2 + 4/5x is sometimes used to model the stopping distance for a car traveling at X miles per hour on dry, level pavement. If a driver can see only 200 ft ahead on a highway with a sharp curve, then safe driving speed X satisfy the quadratic inequality 1/12 x^2+ 4/5x ≤ 200. Solve this inequality to determine safe speed on a curve where a driver can see the road ahead at most 200 ft. What might be a safe speed limit for this curve?

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  1. Since it is not stated in the question, we assume that the unit of f(x) is in feet.

    There is inconsistency with the two expressions for f(x), namely
    f(x)=(1/2)x²+(4/5)x, and later
    f(x)=(1/12)x²+(4/5)x

    Solving the first. If the second expression is intended, you can solve similarly as below:

    f(x)=(1/2)x²+(4/5)x ≤ 200
    (1/2)x²+(4/5)x-200 ≤0
    Using quadratic formula:
    a=1/2
    b=4/5
    c=-200
    x=-20.8 or +19.2 mi/hr
    Reject negative root, so
    x=19.2 mi/hr.

    To stop within 200 feet, the speed must be ≤ 19.2 mi/hr.

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    posted by MathMate

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