Final answer:
To find the height of the cable 48 meters from the center of a parabolic suspension bridge, we establish a coordinate system and use the properties of a parabola. We plug in the coordinates to find the height to be 36 meters.
Step-by-step explanation:
To find the height of the cable at a point 48 meters from the center of a suspension bridge shaped like a parabola, we need to use the properties of a parabolic curve. Given that the towers are 144 meters apart and each is 54 meters high, we can determine the vertex of the parabola is in the center between the towers and at the maximum height of 54 meters.
Let's place the vertex of our parabola at the origin of a coordinate system (0, 54). Since we know that the towers, which are the endpoints of the parabola, are 144 meters apart, they would be located at (-72, 0) and (72, 0). The parabolic equation can be expressed in the form y = ax^2 + c. Here, c is the height of the vertex, so c = 54, and we need to find a using the coordinates of the towers.
Using a tower's coordinates, we get 0 = a*(-72)^2 + 54, leading to a = -54/5184. Therefore, the equation of our parabolic cable is y = (-54/5184)*x^2 + 54.
Now, to find the height of the cable 48 meters from the center, we substitute x = 48 into our equation and calculate y, which is the height of the cable at that point.
y = (-54/5184)*(48)^2 + 54
After computing, we find that the height of the cable 48 meters from the center is 36 meters.