A parabolic suspension bridge cable is hung between two supporting towers 120 meters apart and 35 meters above the bridge deck. The lowest point of the cable is 5 meters above the deck. Determine the lengths of the tension members 20 meters and 40 meters from the bridge center.

To determine the lengths of the tension members, we need to find the equation of the parabola that represents the shape of the suspension bridge cable. We can use the vertex form of a parabolic equation: y = a(x - h)^2 + k.

Given:
- Two supporting towers 120 meters apart. This means the bridge deck spans a distance of 120 meters.
- The lowest point of the cable is 5 meters above the deck.
- The two tension members are located 20 meters and 40 meters from the center (60 meters from each tower).

We can start by finding the vertex (h, k) of the parabola.

The vertex of a parabola can be found by using the formula h = -b / (2a), where h is the x-coordinate of the vertex, and b is the coefficient of the x term in the equation.

In this case, since the parabola is symmetric and the center of the bridge is the vertex, the center is at x = 0. Therefore, h = 0.

Next, we need to find k, the y-coordinate of the vertex. From the given information, we know that the lowest point of the cable is 5 meters above the deck. So, k = 5.

The equation of the parabola now becomes y = ax^2 + 5.

To find the value of a, we can use one of the given points on the parabola, specifically one of the towers. We'll choose the tower on the left side for convenience.

Using the coordinates of the tower (60, 35), we can substitute these values into the equation:

35 = a(60)^2 + 5

Simplifying, we have:

35 = 3600a + 5

Subtracting 5 from both sides, we get:

30 = 3600a

Dividing both sides by 3600:

a = 30 / 3600

Simplifying further, we have:

a = 1 / 120

Now that we have the value of a, the equation of the parabola becomes:

y = (1 / 120)x^2 + 5.

To find the lengths of the tension members 20 meters and 40 meters from the center, we substitute the x-values into the equation and solve for y.

For the tension member 20 meters from the center (x = -20):

y = (1 / 120)(-20)^2 + 5

Simplifying, we have:

y = (1 / 120)(400) + 5

y = 10 / 3 + 5

y = 25 / 3

The length of the tension member 20 meters from the center is 25 / 3 meters.

For the tension member 40 meters from the center (x = 40):

y = (1 / 120)(40)^2 + 5

Simplifying, we have:

y = (1 / 120)(1600) + 5

y = 40 / 3 + 5

y = 55 / 3

The length of the tension member 40 meters from the center is 55 / 3 meters.