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Question 237815: I am given the length, and height of a parabola which is positioned like a bridge, therefore the parabola's opening is downwards. The Length of the base of the parabola is 5meters, the height is 4meters. I need to find the maximum height of a rectangle to fit into the parabola(bridge). The rectangle cannot be the height of the parabola, and the edges cannot go past the parabola. The Width of the rectangle is 2meters. What is the maximum height of the rectangle so that it would fit through the parabola, without getting stuck? How would you write the equation of the parabola? Would it be y= -a(x-4)^2+0 ?? Is the vertex (0,4) ?
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! I am given the length, and height of a parabola which is positioned like a bridge, therefore the parabola's opening is downwards.
The Length of the base of the parabola is 5meters, the height is 4meters.
:
You can use the form y = ax^2 + bx + c to find the parabola
The parabola goes thru the origin, so c = 0, just find a and b
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The vertex coordinates: 2.5, 4; width is given as 5, axis of symmetry will be half this
:
x = 2.5, y = 4
2.5^2a + 2.5b = 4
6.25a + 2.5b = 4; one equation
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and the width of the parabola, the x intercept
x = 5, y = 0
5^2a + 5b = 0
25a + 5b = 0
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use these two equations to solve by elimination
6.25a + 2.5b = 4
25a + 5b = 0
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Multiply the 1st equation by 2
12.5a + 5b = 8
25a + 5b = 0
---------------Subtraction eliminates b, find a
-12.5a = 8
a = 
a = -.64
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Find b using 25a + 5b = 0
25(-.64) + 5b = 0
-16 + 5b = 0
5b = 16
b = 
b = 3.2
:
The equation: y = -.64x^2 + 3.2x
Looks like this
You can see the vertex is 2.5, 4
then
I need to find the maximum height of a rectangle to fit into the parabola(bridge).
The rectangle cannot be the height of the parabola, and the edges cannot go past the parabola.
The Width of the rectangle is 2meters.
What is the maximum height of the rectangle so that it would fit through the parabola, without getting stuck?
:
The rectangle 2 meter width will be centered at x=2.5. therefore
one side will be x=1.5 and other side will be x=3.5
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Find the values for y when x = 1.5 and 3.5, (they will be the same)
y = -.64x^2 + 3.2x
Substitute 1.5 for x and find y
y = -,64(1.5^2) + 3.2(1.5)
y = -1.44 + 4.8
y = 3.36 meters, rectangle height will have to be slightly less than 3.36 meters to fit under the bridge
:
You can check this by finding y when x = 3.5
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Did this make sense to you?
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