Answer:
The vertex of the parabola is (105 , 7)
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)