Question 547162
A concrete bridge over a river has an underside in the shape of a parabolic arch. At the water level, the arch is 20m wide. It has a maximum height of 10m above the water. The minimum vertical thickness of the concrete is 1.5m.
:
I am not sure what the thickness of the concrete means to this problem, anyway, this is an equation for the underside of the arch
:
a) find an algebraic relation that represents the shape of the arch.
three ordered pairs
x=0, y=0; the left side of the arch at the waterline
x=10,y=10; the max height of the arch (the vertex)
x=20,y=0; the right side of the arc at the waterline
:
Using the form ax^2 + bx + c = y
c = 0, so we just need two equations to find a and b
:
x=10, y=10
a*10^2 + 10b = 10
100a + 10b = 10
:
x=20, y=0
a*20^2 + 20b = 0
400a + 20b = 0
Multiply the 1st equation by 2, subtract from the above equation
400a + 20b = 0
200a + 20b = 20
----------------subtraction eliminates b, find a
200a = -20
a = -20/200
a = -.1
;
Use the 1st equation to find b
100(-.1) + 10b = 10
-10 + 10b = 10
10b = 10 + 10
10b = 20
b = 20/10
b = 2
:
The equation for this arch: y = -.1x^2 + 2x
:
Graphically
{{{ graph( 300, 200, -6, 22, -6, 14, -.1x^2+2x) }}}