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Question 1075141: The owner of an animal farm who happens to be an architect is trying to design a parabolic arch for the entrance to his farm. He wants the highest point of the arch to be 40 IN. from the ground. Also included in his design is that at the height of 30 IN. the width of the arch is to be 15 IN. How wide does he want the arch at the ground level? What is the equation of the parabolic arch?
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Your last question cannot be answered as posed. You ask, "What is the equation of the parabolic arch?" There are an infinite number of equations that would describe such a parabola, most of which are impractical for the purposes of computation of the first part of your question, but several of which are, indeed, suitable for determining the width of the arch at ground level.
The reason that there are an infinite number of equations is that even if you restricted the vertex of the parabola to have an ordinate of 40 so that "ground level" would be represented by the  -axis, the abscissa of the vertex could still be any real number. Given that restriction, there are really only three computationally convenient choices for the  -intercept, namely ) with the vertex in QI, ) , or with the vertex in QII.
I chose the first option.
With the information given and the assumption that the vertex of the parabola is at a height of 40 and the parabola intercepts the -axis at the point , we proceed as follows:
Since the width of the parabola at a height of 30 must be 15, there must be a second point on the parabola at ) . Also, since the parabola is symmetric about its axis which passes through the vertex, the abscissa of the vertex, in this case, must be half-way between the two points with function value of 30, namely at )
The general function for a parabola is \ =\ ax^2\ +\ bx\ +\ c) . Using the idea that an ordered pair on the parabola is \right)) , we can establish the following three relationships based on the coordinates of the three given points:
^2\ +\ b(0)\ +\ c\ =\ 30)
^2\ +\ b(7.5)\ +\ c\ =\ 40)
^2\ +\ b(15)\ +\ c\ =\ 30)
Since it is clear from the first equation that  , the 3X3 system simplifies to the following 2X2 system:
All that needs to be done from here is to solve the 2X2 system of linear equations to get the values of the  and  coefficients (you already know the value of  ) then form the function. Then set the function equal to zero and, using the quadratic formula to find the two roots of the equation. The distance between the two base points will be the difference between the roots.
 .
The entrance to a farm that is 3 feet 4 inches tall. Fascinating. Perhaps it is an ant farm.
John
My calculator said it, I believe it, that settles it
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