SOLUTION: A bridge is built in the shape of a parabolic arch. The bridge has a span of 120 feet and a maximum height of 25 feet. Choose a suitable rectangular coordinate system and find the

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Question 79138: A bridge is built in the shape of a parabolic arch. The bridge has a span of 120 feet and a maximum height of 25 feet. Choose a suitable rectangular coordinate system and find the height of the arch at distances of 10, 30 and 50 feet from the center.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A bridge is built in the shape of a parabolic arch. The bridge has a span of 120 feet and a maximum height of 25 feet. Choose a suitable rectangular coordinate system and find the height of the arch at distances of 10, 30 and 50 feet from the center.
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Here's one way to do it:
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We know that equation of a parabola is; ax^2 + bx + c = y
This parabola originates at 0,0 so the value of c = 0; find the values of a & b
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The coordinates of the vertex are given; x/y = 60, 25
60^2*a + 60b = 25
3600a + 60b = 25
Simplify; divide by 5
720x + 12b = 5
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The coordinates of the x intercept: x/y = 120, 0
120^2a + 120b = 0
14400a + 120b = 0
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Multiply the 1st equation by 10 and subtract it from the 2nd equation, find a:
14400a + 120b = 0
7200a + 120b = 50
------------------subtract
7200a + 0b = -50
a = -50/7200
a = -.006944
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Find b: substitute -.006944 for a in the 2nd equation:
14400(-.006944) + 120b = 0
-100 + 120b = 0
120b = +100
b = 100/120
b = +.8333
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The equation of this parabola: y = -.006944x^2 + .8333x
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A look at the graph of this:
+graph%28+300%2C+200%2C+-10%2C+125%2C+-10%2C+30%2C+-.0069x%5E2+%2B+.83x%29+
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"find the height of the arch at distances of 10, 30 and 50 feet from the center.
Substitute the x values in the equation to find the y values
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10 feet from the center would be x=50 and x=70; this will check our equation
they should be equal
x | y
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50|24.3
70|24.3
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30 feet from the center: x=30, x=90
30|18.75
90|18.75
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50 ft from center: x=10, x=110
10|7.64
110|7.64
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Did all this make sense to you?