The cable suspension bridge hangs in the shape of a parabola. The towers supporting the cable area is 600 lft apart and 200ft high. If the cable, at its lowest, is 40ft above the bridge at its midpoint, how high is the cable 50ft away(horizontally) from either tower?
-I’m confused pls help
Let's put all that information on the x-y grid
with the vertex at (0,40)
then the top of the towers are at (-300,200) and (300,200)
then h = a(x-0)^2 + 40 = ax^2 + 40
but (300,200) also lies on it, so
200 = a(300)^2 + 40
a = 160/90000 = 2/1125
height = (2/1125)x^2 + 40
"how high is the cable 50ft away(horizontally) from either tower?"
you want x = 250
find h
btw, the cables on most suspension bridges hang according to a catenary, not a parabola.
Google catenary.
To find the height of the cable 50 feet away from either tower, we can use the concept of a parabolic shape and apply some basic geometry.
Let's start by visualizing the problem. We have two towers 600 feet apart and 200 feet high, and the shape of the cable between the towers follows a parabolic curve. At the midpoint of the cable, it is 40 feet above the bridge.
Given that the cable follows a parabolic shape, we know that the equation of a parabola is of the form y = ax^2 + bx + c. To determine the specific equation for this parabola, we need additional information.
Since the midpoint of the bridge is the vertex of the parabolic curve, we can find the value of 'c' (the constant term in the equation) by substituting the coordinates of the vertex into the equation. In this case, the midpoint is (0, 40), so we have:
40 = a(0)^2 + b(0) + c
40 = c
Therefore, we have the equation of the parabola as y = ax^2 + bx + 40.
To find the values for 'a' and 'b', we need more information. We know that the towers supporting the cable are 600 feet apart and 200 feet high. This means that at x = -300 (to the left of the midpoint) and x = 300 (to the right of the midpoint), the cable touches the tower tops.
Substituting these coordinates into the equation, we can solve for 'a' and 'b'.
When x = -300:
200 = a(-300)^2 + b(-300) + 40
When x = 300:
200 = a(300)^2 + b(300) + 40
Solve these two equations simultaneously to find the values of 'a' and 'b'. Once you have the equation of the parabola, you can substitute x = 50 into the equation to find the height of the cable 50 feet away from either tower.
To find the height of the cable at a point 50ft away from either tower, we can consider the symmetry of the parabolic shape.
Let's first label some points for clarity:
- Tower A: Located at one end, with a height of 200ft.
- Tower B: Located at the other end, also with a height of 200ft.
- Midpoint: The midpoint between the two towers, where the cable is 40ft above the bridge.
Since the cable hangs in the shape of a parabola, it is symmetrical about the midpoint. This means that the height of the cable 50ft away from either tower is the same as the height of the cable 50ft away from the opposite tower.
To find the height of the cable 50ft away from Tower A, we can consider the triangle formed by Tower A, the midpoint, and the point 50ft away from Tower A. Let's call this point C.
The distance between Tower A and the midpoint is given as 300ft, i.e., half of the distance between the two towers. The height of the midpoint (in relation to the bridge) is given as 40ft.
Now, using the properties of a parabola, we can determine the height of the cable at point C:
1. Find the equation of the parabolic shape:
Since it is symmetrical, we can express the equation in vertex form: y = a(x - h)^2 + k, where (h, k) is the vertex.
Using the point (300, 40) as the vertex, we have h = 300 and k = 40.
2. Determine the equation of the parabola:
Given that point (0, 200) lies on the parabola (Tower A's height), we can substitute these values into the equation to find the value of a.
200 = a(0 - 300)^2 + 40
200 = a(90000) + 40
200 - 40 = 90000a
160 = 90000a
a = 160/90000
Therefore, the equation of the parabolic shape is: y = (160/90000)(x - 300)^2 + 40.
3. Determine the height of the cable at point C:
Plug in the x-coordinate of point C (50) into the equation and solve for y:
y = (160/90000)(50 - 300)^2 + 40
y = (160/90000)(-250)^2 + 40
y = (160/90000)(62500) + 40
y = 100 + 40
y = 140
Hence, the height of the cable 50ft away from either tower is 140ft.