You live under a bridge that goes over a river. the underneath side of the bridge is an arc that can be modeled with the function y=-0.000407x^2+0.658x.Where x and y are in feet. How high is the bridge? How long is the section of bridge above the arc?
the height is the y-value at the vertex, where x = .658/.000814
Since y = x(.658-.000407x)
the roots are at 0 and 1616.7
The bridge spans that distance
To determine the height of the bridge, we need to find the maximum point on the arc, which corresponds to the vertex of the quadratic function. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a and b are the coefficients of the quadratic function.
In this case, the quadratic function is y = -0.000407x^2 + 0.658x. Comparing this to the general form of a quadratic equation y = ax^2 + bx + c, we can identify a = -0.000407 and b = 0.658.
Using the formula x = -b / (2a), we can substitute the values for a and b:
x = -0.658 / (2 * -0.000407)
Simplifying the expression:
x = 805.37
Now that we have the x-coordinate of the vertex, we can substitute it back into the equation y = -0.000407x^2 + 0.658x to find the corresponding y-coordinate (the height of the bridge):
y = -0.000407(805.37)^2 + 0.658(805.37)
Calculating the expression:
y ≈ 216.99
Therefore, the height of the bridge is approximately 216.99 feet.
Now, to determine the length of the section of the bridge above the arc, we need to find the x-intercepts of the quadratic function. These points correspond to the locations where the function intersects the x-axis.
To find the x-intercepts, we set y = 0 and solve for x in the equation -0.000407x^2 + 0.658x = 0. This equation can be factored as -0.000407x(x - 1613.75) = 0.
So, we have two solutions:
x = 0
x - 1613.75 = 0
Simplifying the second equation:
x = 1613.75
Therefore, the section of the bridge above the arc has a length of 1613.75 feet.