The cable of a suspension bridge has supporting towers which are 50 ft high and 400 ft apart and is in the shape of a parabola.If the lowest point of the cable is 10 ft above the floor of the bridge, find the length of a supporting rod 100 ft from the center of the span.
Since the vertex is at (0,10), we know that
y = ax^2 + 10
Now, since y(200) = 50,
40000a + 10 = 50
a = 1/1000
at x=50,
y = 1/1000 * 2500 + 10 = 12.5
rats. at x=100,
y = 1/1000 * 100^2 + 10 = 20
Why did the suspension bridge go to therapy?
Because it had trouble finding its lowest point!
But don't worry, I'm here to help you calculate the length of the supporting rod. Using some parabolic fun, let's get to it!
First, let's find the equation of the parabolic cable. We can use the vertex form of a parabola: y = a(x-h)^2 + k, where (h,k) is the vertex.
In this case, the lowest point of the cable is 10 ft above the floor of the bridge, so the vertex is (0, 10).
Now, let's find the equation of the parabola. We know that the supporting towers are 50 ft high, and 400 ft apart. This means that the highest point of the cable is 50 ft above the floor of the bridge, and has a vertex at (200, 50).
Plugging these values into the vertex form equation, we get:
y = a(x-200)^2 + 50
Since the lowest point of the cable is 10 ft above the floor, we can plug in the vertex into the equation:
10 = a(0-200)^2 + 50
Simplifying this, we find:
10 = 40000a + 50
Subtracting 50 from both sides, we get:
-40 = 40000a
Dividing both sides by 40000, we get:
a = -0.001
Now, we can find the length of the supporting rod 100 ft from the center of the span. Let's call this point (x, y).
Substitute x = 100 into the equation of the cable:
y = -0.001(100-200)^2 + 50
Simplifying this, we get:
y = -0.001(10000) + 50
y = -10 + 50
y = 40
So, at a distance of 100 ft from the center, the height of the cable is 40 ft.
Finally, to find the length of the supporting rod, we can use the Pythagorean theorem. The rod forms a right triangle with the floor of the bridge and the cable.
The length of the rod is the hypotenuse, and the height of the cable (40 ft) is one of the legs. The other leg is the distance from the floor to the lowest point of the cable (10 ft).
Using the Pythagorean theorem:
rod length^2 = 40^2 + 10^2
rod length^2 = 1600 + 100
rod length^2 = 1700
Taking the square root of both sides, we find:
rod length ≈ 41.23 ft.
Therefore, the length of the supporting rod 100 ft from the center of the span is approximately 41.23 ft.
To find the length of a supporting rod 100 ft from the center of the span, we need to determine the equation of the parabola representing the shape of the cable.
Let's set up a coordinate system with the origin at the midpoint between the two supporting towers. The y-coordinate will represent the height of the cable above the floor of the bridge.
Since the cable forms a parabolic shape, we can express it in the form of a quadratic equation: y = ax^2 + bx + c.
We know the following points on the parabola:
(200, 50) - the tower on the right
(0, 10) - the lowest point of the cable
(-200, 50) - the tower on the left
Using these points, we can set up a system of equations to solve for the coefficients a, b, and c:
1. 50 = a(200)^2 + b(200) + c
2. 10 = a(0)^2 + b(0) + c
3. 50 = a(-200)^2 + b(-200) + c
Simplifying these equations, we get:
1. 50 = 400a + 200b + c
2. 10 = c
3. 50 = 400a - 200b + c
Since equation 2 tells us that c = 10, we can substitute this into equations 1 and 3:
1. 50 = 400a + 200b + 10
3. 50 = 400a - 200b + 10
Simplifying further, we get:
1. 400a + 200b = 40
3. 400a - 200b = -40
Adding equations 1 and 3 eliminates the b term:
800a = 0
a = 0
Substituting this value of a back into either equation, we get:
400a + 200b = 40
200b = 40
b = 40/200
b = 0.2
Therefore, the equation of the parabola representing the cable is y = 0.2x^2 + 10.
To find the length of the supporting rod 100 ft from the center of the span, we need to determine the height of the cable at that point.
Plugging x = 100 into the equation, we get:
y = 0.2(100)^2 + 10
y = 0.2(10,000) + 10
y = 2,000 + 10
y = 2,010
So, the height of the cable at a distance of 100 ft from the center of the span is 2,010 ft.
To find the length of the supporting rod, we can use the Pythagorean theorem. The supporting rod creates a right-angled triangle, with the height of the cable as one side and the distance from the center of the span as the other side.
Using the formula for the length of a hypotenuse in a right triangle:
Length of the supporting rod = √(height of the cable)^2 + (distance from the center of the span)^2
Substituting the values we have:
Length of the supporting rod = √(2,010)^2 + (100)^2
Length of the supporting rod = √(4,040,100 + 10,000)
Length of the supporting rod = √4,050,100
Length of the supporting rod ≈ 2,012.47 ft
Therefore, the length of the supporting rod 100 ft from the center of the span is approximately 2,012.47 ft.
To find the length of a supporting rod 100 ft from the center of the span, we need to determine the height of the cable at that point.
Let's first set up a coordinate system. We can assume that the midpoint of the span is at the origin (0, 0), and the two supporting towers are located at (-200, 50) and (200, 50).
Since the lowest point of the cable is 10 ft above the floor of the bridge, we can express the equation of the parabolic cable in vertex form. Let the equation be y = a(x - h)^2 + k, where (h, k) is the vertex.
We know that the vertex of the parabola is at the lowest point, so the vertex is (0, 10). Substituting these values into the equation, we get y = a(x - 0)^2 + 10, which simplifies to y = ax^2 + 10.
Now, let's plug in the coordinates of one of the supporting towers to find the value of a. We use (200, 50) in this example, so we have 50 = a(200^2) + 10.
Solving the equation, we get:
50 = 40000a + 10
40000a = 40
a = 40 / 40000
a = 0.001
So, the equation of the cable is y = 0.001x^2 + 10.
To find the height of the cable at a distance of 100 ft from the center of the span, we substitute x = 100 into the equation:
y = 0.001(100^2) + 10
y = 0.001(10000) + 10
y = 10 + 10
y = 20 ft
Therefore, the height of the cable at a point 100 ft from the center of the span is 20 ft.
To find the length of the supporting rod, we need to calculate the length of the vertical segment from the floor of the bridge to the cable at this point.
Using the Pythagorean theorem, the length of the supporting rod is the hypotenuse of a right triangle with the base (horizontal distance) of 100 ft and the height of 20 ft.
The length of the supporting rod can be calculated as:
length = √(base^2 + height^2)
length = √(100^2 + 20^2)
length = √(10000 + 400)
length = √10400
length ≈ 101.98 ft
Therefore, the length of the supporting rod 100 ft from the center of the span is approximately 101.98 ft.