Question 426962: Kayli wants to build a parabolic bridge over a stream in her backyard. The
bridge must span a width of 200 cm. It must be at least 51 cm high where it is
30 cm from the bank on each side. How high will her bridge be?
I have tried many ways of figuring this question out. However, I was
unsuccessful.
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Kayli wants to build a parabolic bridge over a stream in her backyard. The
bridge must span a width of 200 cm. It must be at least 51 cm high where it is
30 cm from the bank on each side. How high will her bridge be?
..
Start by placing the origin of the (x,y) coordinate system at the left bank of the 200 cm span. We now have 3 points of the parabola to work with, the origin,(0,0), 30 cm from the bank,(30,51) and the vertex,(100,k),k(height of bridge) to be determined.Standard form of a parabola: y=A(x-h)^2+k where (h,k) are the (x,y) coordinates of the vertex.
..
Solving for A with two equations:
Using point (30,51)
51=A(30-100)^2+k
Using origin (0,0)
0=A(0-100)^2+k
..
51=A(70)^2+k
0=A(100)^2+k
51=A(4900)+k
0=A(10000)+k
subtract algebraically
51=A(4900-10000)+0
51=A(-5100)
A=51/-5100=-1/100
solving for k
y=-1/100(x-100)^2+k
51=-1/100(30-100)^2+k
51=-49+k
k=100 cm
ans:height of parabolic bridge=100cm
See graph of parabola below:
y=-(1/100)(x-100)^2+100
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