Question 1155433
<br>
Let the center of the arch at ground level be the origin of a coordinate system.  Then the parabola passes through the points (-92,0), (0,40), and (92,0).<br>
With the y-intercept at (0,40), the equation is of the form {{{y = -ax^2+40}}}.<br>
Use the point (92,0) to determine the value of a to complete the equation.<br>
{{{0 = -a(92^2)+40}}}
{{{a(92^2) = 40}}}
{{{a = 40/(92^2)}}}
<br>
The equation of the parabola is<br>
{{{y = -(40/92^2)x^2+40}}}<br>
{{{graph(400,400,-100,100,-10,50,-(40/92^2)x^2+40)}}}<br>
Evaluate y when x=15 to find the answer to the problem.<br>
{{{-(40/92^2)(15^2)+40}}} = 38.936673 to several decimal places.<br>
Here is a way to find this value without finding the equation of the parabola.<br>
The basic form of the equation of a parabola is {{{y = ax^2}}}.<br>
This equation tells us directly that the vertical displacement is proportional to the square of the horizontal displacement.<br>
We know, from the points (0,40) and (92,0) on the parabola, that the vertical displacement is -40 for a horizontal displacement of 92.<br>
The question asks for the height of the bridge 15 feet from the center; we can find that height by determining the vertical displacement from the vertex at (0,40) for a horizontal displacement of 15 from the vertex.<br>
The ratio of horizontal displacements for the two points is 15/92; the ratio of the vertical displacements will be the square of that.  So the vertical displacement 15 feet from the center of the arch is<br>
{{{(15/92)^2(-40) = -1.063327}}}<br>
and so the height of the arch 15 feet from its center is 40-1.063327 = 38.936673 feet.<br>