MATH HELP PLEASEEEEEEEE!!!!

Question

A bridge with a parabolic inner arch is being constructed over a canal. The arch will just touch the water on each side of the canal, which is 27.2 m wide. The arch must have a clearance of 6.4 m at a horizontal distance of 9.5 m from either bank of the canal. a) Write a relation, in factored form, to model the height, h, in metres, of the bridge at a horizontal distance of x metres from one bank of the canal. b) Rewrite your relation for part a) in vertex form by expanding and completing the square. c) How high above the water will the arch be, at its highest point, to the nearest hundredth of a metre?

Answer 1

(9.5, 6.4), (17.7, 6.4), (27.2, 0), (0.0)

are points on the downward opening parabola

y=a(x-d)^2 +k in vertex form

axis of symmetry is: x=27.2/2 = 13.6

13.6 = x coordinate of the vertex=d

(13.6, k)= vertex, k=max height

a<0

h=height

h(x)= a(x-27.2)(x-0)

plug in a point to solve for "a"

6.4= a(9.5-27.2)(9.5)

a = 6.4/(-17.7)(9.5)=-.038061254

h = ((6.4)/(-17.7)(9.5))(x^2 -27.2x) in factored form

complete the square

h=-((6.4)/(17.7)(9.5))(x^2-27.2x + (13.6)^2) + (6.4)(13/6)^2/(17.7)(9.5)

h(x)=-.038061254(x-13.6)^2 + 7.039809694 in vertex form

max height = the constant term

= 7.04 meter, rounded to nearest hundredth