You live near a bridge that goes over a river. The underside of the bridge is an arch that can be modeled with the function y=-0.000475x^2+0.851x, where x and y are in feet. How high above the river is the bridge(the top of the arch)? How long is the section of bridge above the arch? How long is the section of bridge above the arch?
y = x(-.000475x+.851)
y=0 at x=0,1791
The vertex is midway between the roots
So, at x=896, y=381
The length of the bridge is the distance between the roots. (assuming the bridge just touches the river banks at the water's edge -- not too likely in the real world)
To find the height of the bridge above the river, you need to determine the maximum point on the arch. The maximum point of a quadratic function can be found using the vertex formula. The vertex formula for a quadratic function in the form y = ax^2 + bx + c is given by:
x = -b / (2a)
In this case, the function describing the arch is y = -0.000475x^2 + 0.851x. Comparing this with the general form of a quadratic function, we can determine that a = -0.000475 and b = 0.851. Plugging these values into the vertex formula, we have:
x = -0.851 / (2 * -0.000475)
Simplifying, we get:
x ≈ 897.895
Now, substitute this value back into the original equation to find the y-coordinate of the maximum point:
y ≈ -0.000475(897.895)^2 + 0.851(897.895)
Simplifying, we get:
y ≈ 381.754
Therefore, the bridge is approximately 381.754 feet high above the river at its highest point.
To find the length of the section of the bridge above the arch, we need to determine the x-values where the function y = -0.000475x^2 + 0.851x intersects with the x-axis. These represent the endpoints of the section above the arch. To find the x-intercepts, we set y = 0 and solve for x:
0 = -0.000475x^2 + 0.851x
Simplifying, we have:
0.000475x^2 - 0.851x = 0
Factor out x:
x(0.000475x - 0.851) = 0
Setting each factor equal to zero, we have:
x = 0 (extraneous solution)
0.000475x - 0.851 = 0
Solving for x, we get:
x ≈ 1789.474
Therefore, the section of the bridge above the arch has a length of approximately 1789.474 feet.