Question 85825
The problem tells you that the equation for the height of the ball is:
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{{{h = -16t^2 + 48t + 2}}}
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It asks you to find the time that the ball reaches a height of 38 feet.
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Substitute 38 feet into the equation. Since 38 feet is the height, it replaces h in the
equation, and the equation becomes:
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{{{38 = -16t^2 + 48t + 2}}}
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The standard form for solving equations of this type is:
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{{{at^2 + bt + c = 0}}}
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in which a, b, and c are integers.
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You can work the height equation into this form. Begin by transposing (switching sides) of
the height equation to get:
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{{{-16t^2 + 48t + 2 = 38}}}
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Next subtract 38 from both sides of the equation:
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{{{-16t^2 + 48t + 2 - 38 = 0}}}
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and after the subtraction the left side becomes:
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{{{ -16t^2 + 48t -36 = 0}}}
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Notice that this equation now matches the standard form ... in which a = -16, b = +48, and
c = -36.
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You can simplify this a little by noting that 4 is a factor of all the terms. So divide
all the terms in this equation by 4 to get:
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{{{-4t^2 + 12t - 9 = 0}}}
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You can further simplify this by multiplying all the terms on both sides by -1 to get:
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{{{4t^2 - 12t + 9 = 0}}}
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This can be solved by using the quadratic formula, or you might notice that the left side
of this equation factors to give:
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{{{(2t -3)^2 =  0}}}
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The equation will be true when the left side equals zero.  Therefore, it will be true if
{{{2t - 3 = 0}}}
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Solve this equation by adding 3 to both sides to get:
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{{{ 2t = 3}}}
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and solve for t by dividing both sides by 2 to get:
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{{{t = 3/2 = 1.5 }}} seconds.
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So the ball reaches a height of 38 feet 1.5 seconds after Tonisha hits it.
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Hope this helps you to understand the problem.