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?
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.
To find the height of the bridge above the river (the top of the arch), we need to find the maximum value of the function y = -0.000475x^2 + 0.851x.
Step 1: To find the maximum value, we need to determine the x-coordinate of the vertex of the function. The x-coordinate can be found using the formula x = -b / (2a), where a and b are the coefficients of x^2 and x respectively.
Comparing the equation y = -0.000475x^2 + 0.851x to the general quadratic equation form y = ax^2 + bx + c, we can identify that a = -0.000475 and b = 0.851.
Using the formula x = -b / (2a):
x = -0.851 / (2 * -0.000475)
x = 1789.47
Step 2: Next, we substitute the x-coordinate of the vertex back into the original equation to find the corresponding y-coordinate.
y = -0.000475(1789.47)^2 + 0.851(1789.47)
y ≈ 756.95
The height of the bridge above the river (the top of the arch) is approximately 756.95 feet.
Now, to find the length of the section of the bridge above the arch, we need to determine the x-values where the function intersects the x-axis.
Step 3: Set y = 0 in the original equation and solve for x:
0 = -0.000475x^2 + 0.851x
Using factoring or the quadratic formula, we find the x-values:
x1 ≈ 0
x2 ≈ 1789.47
Step 4: Finally, the length of the section of the bridge above the arch is the difference between the two x-values:
Length = x2 - x1
Length ≈ 1789.47 - 0
Length ≈ 1789.47 feet
The section of the bridge above the arch is approximately 1789.47 feet long.
To find the height of the bridge above the river (the top of the arch), we need to find the maximum value of the function y = –0.000475x^2 + 0.851x. This can be done by finding the vertex of the parabolic function.
The vertex of a parabola can be found using the formula x = -b/2a, where a and b are the coefficients of the quadratic function in the form ax^2 + bx + c.
In this case, a = -0.000475 and b = 0.851. Plugging these values into the formula, we get:
x = -0.851 / (2 * (-0.000475))
x = 1788.42
So, the x-coordinate of the vertex is approximately 1788.42.
To find the corresponding y-coordinate of the vertex, we substitute this value back into the equation:
y = –0.000475(1788.42)^2 + 0.851(1788.42)
y ≈ 760.82
Therefore, the height of the bridge above the river (the top of the arch) is approximately 760.82 feet.
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. These represent the points where the arch ends and the bridge starts.
To find these x-intercepts, we set y = 0 and solve for x:
0 = –0.000475x^2 + 0.851x
This equation is a quadratic equation, which can be solved using factoring, completing the square, or the quadratic formula. In this case, using the quadratic formula is the most straightforward approach.
The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a).
In our equation, a = -0.000475, b = 0.851, c = 0.
Plugging in these values, we get:
x = (-0.851 ± √(0.851^2 - 4 * (-0.000475) * 0)) / (2 * (-0.000475))
x = (-0.851 ± √(0.72360136875)) / (-0.00095)
Calculating the values inside the square root:
x ≈ -151.80 or x ≈ 1787.423
Therefore, the x-values where the function intersects the x-axis (the ends of the arch) are approximately -151.80 and 1787.423.
To find the length of the bridge above the arch, we subtract the x-coordinate at the left end of the arch from the x-coordinate at the right end of the arch:
Length = 1787.423 - (-151.80)
Length ≈ 1939.223 feet
Therefore, the length of the section of the bridge above the arch is approximately 1939.223 feet.