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 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 381.16 ft.
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. The formula of the arch can be represented as y = -0.000475x^2 + 0.851x.
To find the x-coordinate of the maximum point (vertex) of the parabola, we can use the formula x = -b / (2a), where a is the coefficient of x^2 and b is the coefficient of x.
In this case, a = -0.000475 and b = 0.851.
Using the formula x = -0.851 / (2*(-0.000475)), we find that x ≈ 1788.42.
To find the corresponding y-coordinate of the maximum point, we substitute the x-value into the equation:
y = -0.000475 * (1788.42)^2 + 0.851 * 1788.42 ≈ 1,791.58 ft.
Therefore, the height of the bridge above the river is approximately 1,791.58 ft.
To find the length of the section of the bridge above the arch, we need to find the x-values where the function is equal to zero. These correspond to the points where the bridge meets the arch.
Setting y = 0 in the equation -0.000475x^2 + 0.851x = 0, and solving for x, we get:
-0.000475x^2 + 0.851x = 0
x * (-0.000475x + 0.851) = 0
This equation is satisfied when x = 0 or when -0.000475x + 0.851 = 0.
Solving -0.000475x + 0.851 = 0, we can find the x-value where the bridge intersects the arch.
Rearranging the equation, we get:
-0.000475x = -0.851
x = -0.851 / -0.000475 ≈ 1788.42
Therefore, the bridge intersects the arch at approximately x = 0 and x ≈ 1788.42.
The length of the section of the bridge above the arch is the distance between these two x-values:
Length = 1788.42 - 0 ≈ 1788.42 ft.
Therefore, the length of the section of the bridge above the arch is approximately 1788.42 ft.
So, 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 1788.42 ft.
To determine 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 parabola defined by the function.
The x-coordinate of the vertex can be found using the formula x = -b/(2a), where a and b are the coefficients of x^2 and x, respectively. In this case, a = -0.000475 and b = 0.851. Plugging these values into the formula, we get x = -0.851/(2*-0.000475) = 1788.42.
To find the y-coordinate of the vertex, we substitute the x-coordinate into the function y = -0.000475x^2 + 0.851x. Using x = 1788.42, we get y = -0.000475*(1788.42)^2 + 0.851*(1788.42) = 1,791.58.
Therefore, the bridge is about 1,791.58 ft. above the river.
To determine the length of the section of the bridge above the arch, we need to find the x-values where the function crosses the x-axis. These are the values for which y = 0.
Setting y = 0 in the function -0.000475x^2 + 0.851x = 0, we can solve for x. Factoring out an x, we get x(-0.000475x + 0.851) = 0. This equation is true if either x = 0 or -0.000475x + 0.851 = 0.
From the second equation: -0.000475x + 0.851 = 0, we can solve for x to find the x-intercept. Subtracting 0.851 from both sides and then dividing by -0.000475, we get x ≈ 1787.368.
The length of the section of the bridge above the arch can be found by subtracting the x-coordinate of the vertex (1788.42) from the x-intercept (1787.368). The difference is approximately 381.16 ft.
Therefore, the length of the section of the bridge above the arch is about 381.16 ft.
So 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 381.16 ft.