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 = –0.000475x2 + 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?

Answer 1

To find the height of the bridge, we need to find the maximum value of the function. The function is in the form of a quadratic equation, where the coefficient of x^2 is negative. This means that the highest point of the arch will be 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 x^2 and x terms respectively.
For the given function, the coefficient of x^2 is -0.000475 and the coefficient of x is 0.851. Plugging these values into the formula, we have:
x = -(0.851)/ (2 * (-0.000475))
x = 0.851 / 0.00095
x ≈ 894.74
So, the x-coordinate of the vertex is approximately 894.74.

Now, we need to find the y-coordinate of the vertex, which represents the maximum height of the bridge. We can substitute the x-coordinate of the vertex into the function to find y:
y = -0.000475(894.74)^2 + 0.851(894.74)
y ≈ 678.13

Therefore, the bridge is approximately 678.13 feet above the river, which is the height of the arch.

To find the length of the section of the bridge above the arch, we need to find the x-values where the function intersects the x-axis. In other words, we need to find the roots of the quadratic equation. Since the arch is symmetric, the two roots will have the same magnitude but opposite signs.

To find the roots, we set y = 0 and solve for x:
0 = -0.000475x^2 + 0.851x
0.000475x^2 - 0.851x = 0
x(0.000475x - 0.851) = 0

Setting each factor equal to 0:
x = 0
0.000475x - 0.851 = 0

Solving the second equation:
0.000475x = 0.851
x = 0.851 / 0.000475
x ≈ 1793.68

So, the two roots are approximately x = 0 and x = 1793.68.

Since the arch is symmetric, the length of the section of the bridge above the arch is twice the distance from the vertex to the x-intercept. This can be calculated as:
2 * (1793.68 - 894.74) ≈ 1798.94 feet

Therefore, the length of the section of the bridge above the arch is approximately 1798.94 feet.