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?
(1 point)
Responses
The bridge is about 1,791.58 ft. above the river, and the length of the bridge above the arch is about 381.16 ft.
The bridge is about 1,791.58 ft. above the river, and the length of the bridge above the arch is about 381.16 ft.
The bridge is about 1,791.58 ft. above the river, and the length of the bridge above the arch is about 895.79 ft.
The bridge is about 1,791.58 ft. above the river, and the length of the bridge above the arch is about 895.79 ft.
The bridge is about 381.16 ft. above the river, and the length of the bridge above the arch is about 895.79 ft.
The bridge is about 381.16 ft. above the river, and the length of the bridge above the arch is about 895.79 ft.
The bridge is about 381.16 ft. above the river, and the length of the bridge above the arch is about 1,791.58 ft.
The bridge is about 381.16 ft. above the river, and the length of the bridge above the arch is about 1,791.58 ft.
To find the height of the bridge (the top of the arch), you need to determine the maximum value of the function y = –0.000475x^2 + 0.851x. This can be done by taking the derivative of the function with respect to x, setting it equal to zero, and solving for x.
The derivative of y with respect to x is dy/dx = (-0.00095x + 0.851). Setting dy/dx equal to zero, we get:
-0.00095x + 0.851 = 0
-0.00095x = -0.851
x = -0.851 / -0.00095
x ≈ 895.79
Now, plug this value of x back into the function to find the corresponding y-coordinate:
y = -0.000475(895.79)^2 + 0.851(895.79)
y ≈ -0.000475(802284.0841) + 762.83129
y ≈ -381.16 + 762.83129
y ≈ 381.67129
Therefore, the bridge is about 381.16 ft above the river.
To find the length of the section of the bridge above the arch, you need to calculate the x-values where the function y = –0.000475x^2 + 0.851x intersects the x-axis. This can be done by setting y equal to zero and solving for x.
-0.000475x^2 + 0.851x = 0
x(-0.000475x + 0.851) = 0
From this, we get two solutions:
x = 0 (which does not correspond to the section of the bridge above the arch)
and
-0.000475x + 0.851 = 0
-0.000475x = -0.851
x = -0.851 / -0.000475
x ≈ 1780
Therefore, the length of the section of the bridge above the arch is about 1780 ft.
To find the height of the bridge, we need to find the maximum value of the function y = -0.000475x^2 + 0.851x.
The maximum value of a quadratic function ax^2 + bx + c occurs at the vertex, which is given by the formula x = -b / (2a).
In this case, a = -0.000475 and b = 0.851.
x = -0.851 / (2 * -0.000475)
x = 1786.842
Now, substitute this value of x back into the original equation to find the height of the bridge:
y = -0.000475(1786.842)^2 + 0.851(1786.842)
y ≈ 1,791.58 ft.
So, the bridge is about 1,791.58 ft above the river.
To find the length of the section of the bridge above the arch, we need to calculate the x-values where y = 0 (the bridge touches the river).
Setting the equation equal to zero:
-0.000475x^2 + 0.851x = 0
Factor out x:
x(-0.000475x + 0.851) = 0
We have two solutions:
x = 0 (which corresponds to the starting point of the bridge)
and
-0.000475x + 0.851 = 0
-0.000475x = -0.851
x ≈ 1786.842
So, the length of the section of the bridge above the arch is approximately 1786.842 ft.
To find the height of the bridge above the river (the top of the arch), we need to evaluate the function at the vertex. The vertex of a quadratic function in the form y = ax^2 + bx + c is given by the formula x = -b/2a.
Given the function y = –0.000475x^2 + 0.851x, we can see that a = -0.000475 and b = 0.851. Plugging these values into the formula, we have x = -0.851 / (2 * -0.000475) = 1784.21 ft.
Now we can substitute this x-value into the function to find the corresponding y-value, which represents the height of the bridge above the river: y = –0.000475(1784.21)^2 + 0.851(1784.21) ≈ 1,791.58 ft.
So, the bridge is about 1,791.58 ft. above the river.
To find the length of the section of the bridge above the arch, we need to find the x-values where the function y = –0.000475x^2 + 0.851x intersects the x-axis. These values will give us the limits of the bridge above the arch.
To find the x-intercepts, we set y = 0 and solve the quadratic equation:
0 = –0.000475x^2 + 0.851x
Next, we can factor out the common factor:
0 = x(–0.000475x + 0.851)
Setting each factor equal to zero, we have:
x = 0 (ignoring this solution for our scenario since it represents the point at the very beginning of the bridge)
-0.000475x + 0.851 = 0
Solving for x, we get:
x = 0.851 / 0.000475 ≈ 1,788.42 ft
Since the bridge is symmetrical, the length of the section of the bridge above the arch is twice this value:
Length = 2 * 1,788.42 ≈ 3,576.84 ft
So, the length of the section of the bridge above the arch is about 3,576.84 ft.
Therefore, the correct response is: The bridge is about 1,791.58 ft. above the river, and the length of the bridge above the arch is about 3,576.84 ft.