A horizontal bridge is in the shape of a parabolic arch Give

A horizontal bridge is in the shape of a parabolic arch. Given the information shown in this figure, find the equation of the arch. (Use the form y = ax^2 or y = ax^2 + b) What is the height of the arch 2 feet from the shore?

Solution

Lets assume the parabola : y = ax^2 +bx +c

bridge sitting on the x-axis and centered on the y-axis, then the parabola passes through (-15,0), (0,10), and (15,0) :

plug all three points to find a, b , c: 10 = a*0 +b*0 +c ----> c = 10

0 = 225a +15b +10----> 45a +3b = -2

0 = 225a -15b +10 ----> 45a -3b = -2

we get : 90a = -4 ---> a = -2/45 ; 6b =0 ; b=0

a) y = -2x^2/45 +10

b) height of bridge 2ft from the shore i.e x =13 ft from origin:

y = -2(13)^2/45 +10 = -8/45+10 = 2.49 feet