A bridge forms a parabolic arch. The span of the arch is 60m and its centre is 10m above either end. What equation models this situation?

Question

This is my quadratics unit

Answer 1

Hi Yudy.

 

For this problem it's easiest to draw the y-axis through the peak of the bridge and the x-axis through the base. This will give you a y-intercept of (0,60) and x-intercepts of (-30,0) and (30,0).

 

A parabola has the following equation:

 

y = ax^2 + bx + c

 

We have 3 points that we know:

(0,60)

(30,0)

(-30,0)

 

We can plug these points in to solve for a, b, and c.

 

60 =a(0^2) + b(0) + c

0 = a(30)^2 + b(30) + c

0 = a(-30)^2 + b(-30) + c

 

From equation 1:

c = 60

 

From equations 2 and 3:

0 = 900a + 30b + 60

0 = 900a - 30b + 60

 

Add these two equations together.

 

0 = 1800a + 120

 

-120 = 1800a

 

a = -120/1800

 

a = -1/15

 

Now plug a and c into one of your equations.

 

0 = 900(-1/15) + 30b + 60

 

0 = -60 + 30b + 60

 

b = 0

 

So, your equation will be:

 

y = -1/15x^2 + 60

 

Hope this helps.