SOLUTION: A parabolic arch has a span of 120ft and a maximum height of 25ft. Choose a suitable rectangular system to find the equation of the parabola. Then calculate the height of the arc
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Question 199973: A parabolic arch has a span of 120ft and a maximum height of 25ft. Choose a suitable rectangular system to find the equation of the parabola. Then calculate the height of the arch 10, 20, and 40 ft from the center. Provide a sketch representing the situation and be sure it is clearly labeled with coordinates.
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
A parabolic arch has a span of 120 feet and a maximum height of 25 feet.
Choose a suitable rectangular coordinate system and find the equation of the
parabola. Then calculate the height of the arch 10, 20 and 40 feet from the center.
:
Find the equation using the format y = ax^2 + bx + c (c=0, we can ignore that)
:
Using the vertex
x=60, y = 25
a(60^2) + 60b = 25
3600a + 60b = 25
:
Using the x intercept
x=120, y=0
a(120^2) + 120b = 0
14400a + 120b = 0
:
Multiply the 1st equation by 2, subtract from the 2nd equation
14400a + 120b = 0
7200a + 120b = 25
--------------------subtraction eliminates b, find a
7200a = -25
a =
a = -.00694
:
Find b using the 2nd equation, substitute -.00694 for a
14400(-.00964) + 120b = 0
-100 + 120b = 0
120b = 100
b =
b = .833
:
y = -.00694x^2 + .8333x; the equation for this parabola
:
10 ft from center: 60-10 = 50 = x
y = -.00694(50^2) + .8333(50)
y = -17.35 + 41.665
y = 24.3 ft high, 10 ft from center
:
20 ft from center: 60-20 = 40 = x
y = -.00694(40^2) + .8333(40)
y = -11.1 + 33.3
y = 22.2 ft high. 20 ft from center
:
40 ft from center: 60-40 = 20 = x
y = -.00694(20^2) + .8333(20)
y = -2.8 + 16.7
y = 13.9 ft high. 40 ft from center
:
:
The graph of this equation:
:
Provide a sketch representing the situation and be sure it is clearly labeled with coordinates.
Your sketch should resemble this graph, you should be able to label this now
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