If the distribution of weight is uniform, then the rope of the suspended bridge takes the form of parabola.The height of the supporting towers is 20m, the distance between these towers is 150m and the height of the lowest point of the rope from the road is 3m. Find the equation of the parabolic shape of the rope considering the floor of the parabolic shape of the rope considering the floor of the bridge as X-axis and the axis of the parabola as Y-axis. Find the height of that tower which supports the rope and is at a distance of 30 m from the centre of the road.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Geometry of the Problem
We have a parabolic rope suspended between two towers. The height of the towers is 20 m, the distance between the towers is 150 m, and the lowest point of the rope is 3 m above the road.

### Step 2: Set Up the Coordinate System
Let's set the origin (0, 0) at the lowest point of the rope. Therefore, the coordinates of the towers will be:
- Left tower: (-75, 20) (since the distance from the center is half of 150 m)
- Right tower: (75, 20)

### Step 3: Determine the Vertex of the Parabola
The vertex of the parabola (the lowest point of the rope) is at (0, 3).

### Step 4: Use the Standard Form of the Parabola
The equation of a parabola that opens upwards can be written as:
\[ y = ax^2 + bx + c \]
Since the vertex is at (0, 3), we can rewrite the equation as:
\[ y = a(x^2) + 3 \]

### Step 5: Use the Coordinates of the Towers to Find 'a'
We know that the height of the towers is 20 m, so we can use the coordinates of one of the towers to find 'a'. Let's use the left tower (-75, 20):
\[ 20 = a(-75)^2 + 3 \]
\[ 20 = 5625a + 3 \]
\[ 5625a = 20 - 3 \]
\[ 5625a = 17 \]
\[ a = \frac{17}{5625} \]

### Step 6: Write the Equation of the Parabola
Substituting 'a' back into the equation:
\[ y = \frac{17}{5625}x^2 + 3 \]

### Step 7: Find the Height of the Tower at 30 m from the Center
To find the height of the tower that is 30 m from the center, we substitute \( x = 30 \) into the equation of the parabola:
\[ y = \frac{17}{5625}(30^2) + 3 \]
\[ y = \frac{17}{5625}(900) + 3 \]
\[ y = \frac{15300}{5625} + 3 \]
\[ y = 2.72 + 3 \]
\[ y = 5.72 \]

### Step 8: Calculate the Height of the Tower
The height of the tower at a distance of 30 m from the center is:
\[ \text{Height} = 5.72 \text{ m} \]

### Final Answer
The equation of the parabolic shape of the rope is:
\[ y = \frac{17}{5625}x^2 + 3 \]
The height of the tower at a distance of 30 m from the center is 5.72 m.

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