The main cable of a suspension bridge forms a
parabola, described by the equation
y = a(x - h)2 + k.
0
27
52.5 105
127
157.6
12
210
27
y = height in feet of the cable above the roadway
x = horizontal distance in feet from the left bridge
support
a = a constant
(h, k) = vertex of the parbola
what is the vertex of the parbola?
done

The vertex of the parabola is (105 , 7)

Step-by-step explanation:

* Lets explain how to solve the problem

- The equation of the parabola is y = a(x - h)² + k, where (h , k) are

  the coordinates of the vertex point of the parabola

- The points (0 , 27) , (52.5 , 12) , (105 , 7) ,  (157.6 , 12) , (210 , 27) are

 the points lie on the parabola

- We have three unknown a , h , k to find them we will substitute the x

 and y in the equation by the coordinates of some point on the

 parabola

- Lets start with point (0 , 27)

∵ x = 0 and y = 27

∴ 27 = a(0 - h)² + k

∴ 27 = ah² + k ⇒ (1)

- Lets use point (210 , 27)

∵ x = 210 and y = 27

∴ 27 = a(210 - h)² + k ⇒ (2)

- Equations (1) and (2) have the same L.H.S, so we can equate them

∴ ah² + k = a(210 - h)² + k ⇒ subtract k from both sides

∴ ah² = a(210 - h)² ⇒ divide both sides by a

∴ h² = (210 - h)² ⇒ take √ for both sides

∴ h = ± (210 - h)

∵ h = 210 - h ⇒ add h to both sides

∴ 2h = 210 ⇒ divide both sides by 2

∴ h = 105

∵ h = - (210 - h)

∴ h = -210 + h ⇒ no value of h from this equation so we will ignore it

∴ The value of h is 105

- Lets substitute this value of h in the equation

∴ y = a(x - 105)² + k

- Lets use the point (105 , 7)

∵ x = 105 and y = 7

∴ 7 = a(105 - 105)² + k

∴ 7 = a(0) + k

∴ k = 7

- The coordinates of the vertex point are (h , k)

∵ h = 105 and k = 7

∴ The vertex of the parabola is (105 , 7)

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