Question 479248
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Set your parabola so that the vertex is at (0,8).  That means the floor is the x-axis and the two intercepts are (-5,0) and (5,0).


Let the function *[tex \Large f] that describes this parabola be defined as *[tex \Large f(x)\ =\ ax^2\ +\ bx\ +\ c]


Substituting the coordinate values from the three points we know must be on the parabola:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a(0^2)\ +\ b(0)\ +\ c\ =\ 8]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a(-5^2)\ +\ b(-5)\ +\ c\ =\ 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a(5^2)\ +\ b(5)\ +\ c\ =\ 0]


Simplifying the first equation gives us:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ c\ =\ 8]


Substituting and simplifying the other two equations results in the following 2X2 linear system:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 25a\ -\ 5b\ =\ -8]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 25a\ +\ 5b\ =\ -8]


The solution of the system and therefore the coordinates of the parabolic function is *[tex \Large a\ =\ -0.32\ \ ], *[tex \Large b\ =\ 0\ \ ], and *[tex \Large c\ =\ 8], verification of which I leave as an exercise for the student.


Therefore the desired function is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ -0.32x^2\ +\ 8]


Now if a six-foot tall person walks through the door so that the centerline of the person's body is 2 feet right or left of the centerline of the doorway, does the person hit their head?


If the person's head were infinitely thin, then it would be a simple matter of determining the value of the function at either *[tex \Large x\ =\ \pm{2}].  However, in practical terms, given that the average six-foot tall person has significantly greater cranial volume than the average algebra student, we'll use 6 inches as a rough approximation of the width of the head, half of which is 3 inches.  Hence, we need to determine the function value at *[tex \Large x\ =\ \pm{2.25}]


Again, leaving the verification arithmetic to the student, a 6-foot tall person with a head the shape of a rectangular prism with a width along an axis parallel to the plane of the door of 6 inches would clear the door 2 feet either side of center by approximately 4.5 inches.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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