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?
y = x (.658 - .000407 x)
where is y = 0 ?
at x = 0 and at x = .658/.000407 = 1617 ft long (that is part B)
this is a parabola that crosses the x axis at 0 and 1617
so the vertex, max height, is halfway between
at x = 808 what is the height?
y = -.000407 (808^2) + .658(808)
= 266 feet high (Part A)
correct answer is "The bridge is about 381.16 ft. above the river, and the length of the bridge above the arch is 1,791.58 ft."
To find the height of the bridge, we need to determine the maximum point (or vertex) of the arc. This point will give us the highest point the bridge reaches.
The equation of the arc is y = -0.000407x^2 + 0.658x. Since this is a quadratic equation in the form y = ax^2 + bx + c, we can determine the x-coordinate of the vertex using the formula x = -b/ (2a).
In this case, a = -0.000407 and b = 0.658. Plugging these values into the formula, we have:
x = -0.658 / (2 * -0.000407)
x ≈ 806.62 feet
Now that we know the x-coordinate of the vertex, we can substitute it back into the equation to find the y-coordinate (or height) of the bridge:
y = -0.000407(806.62)^2 + 0.658(806.62)
y ≈ 523.71 feet
Therefore, the height of the bridge is approximately 523.71 feet.
To find the length of the section of the bridge above the arc, we need to determine the x-values where the arc intersects with the x-axis (y = 0). These points will give us the endpoints of the section of the bridge above the arc.
Setting the equation y = -0.000407x^2 + 0.658x equal to 0, we can solve for x:
-0.000407x^2 + 0.658x = 0
Factoring out x, we have:
x(-0.000407x + 0.658) = 0
This equation will be true if either x = 0 (which is not relevant in this case) or -0.000407x + 0.658 = 0. Solving for x in the second equation, we have:
-0.000407x = -0.658
Dividing both sides by -0.000407, we find:
x ≈ 1610.57 feet
Therefore, the section of the bridge above the arc has a length of approximately 1610.57 feet.