SOLUTION: A parabolic arch is constructed which is 8 ft wide at the base and 9 feet tall. Find the height of the arch exactly 1 foot in from the base of the arch

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Question 1091906: A parabolic arch is constructed which is 8 ft wide at the base and 9 feet tall. Find the height of the arch exactly 1 foot in from the base of the arch
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!

Put the top of the arch (the vertex of the parabola) at the origin. Then since the parabola opens downward, the equation of the parabola is of the form
y+=+ax%5E2
where a is a negative constant.

If the arch is 8 feet wide and 9 feet high, then the coordinates of the bases of the arch are (-4,-9) and (4,-9). We can use the coordinates of either of those points to determine the value of the constant a.

-9+=+a%284%29%5E2
-9+=+16a
a+=+-9%2F16

So the equation of the parabola is
y+=+-%289%2F16%29x%5E2

One foot from the right base of the arch, the x value is 3; that makes the y value
-%289%2F16%29%283%5E2%29+=+-81%2F16

The number we want is the height of the arch 1 foot from its base; that is
%28-81%2F16%29-%28-9%29+=+-81%2F16+%2B+144%2F16+=+63%2F16

The height of the arch 1 foot from either base is 63/16 feet, or 3 15/16 feet.