ind the coordinates of the vertex of the parabola
y = 2x2 − 4x + 9 b) One train route follows the curve
y = 2x2 − 4x + 9,
and the other follows y = x2. Assume that distance along both axes is measured in miles. The railroad wants to construct a north-south maintenance road PQ between the two routes. Where should the road be located so that it is as short as possible?
Q(x, y) =
P(x, y) =
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the vertical distance between two curves is given by taking the difference between them ...\[(2x^2 - 4x + 9) - (x^2) = x^2 - 4x + 9 = (x-2)^2 + 5\]Looks like the two curves are 5 apart when x = 2 between (2,9) and (2,4)...
find the line of symmetry of the parabola....x = -b/2a
x = -(-4)/(2x2) line of symmetry for the parabola is x = 1
substitute into th equation to find y Vertex is (1, 7)
the access road will be on the line x = 1/2 mid way between the two lines of symmetry.
find the equation of the line between the two vertices (0,0) and (1, 7)
the equation is y = 7x.... sub x = 1/2 into the equation then Q 1/2, 3 1/2)|dw:1327804161327:dw|
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The requirement is that the road be north-south, so (luckily) we have the same value of x for the points P and Q.
The distance is shortest if y1'(x) = y2'(x).
to solve for x=2.
So P(2,y1(2)), and Q(2,y2(2)) to give the shortest distance as 9-4=5 miles.