A cable is suspended from two towers with a distance of 100m between them. The 1st tower has

a height of 50m and the 2nd tower has a height of 70m from the ground. The lowest part of the
cable has a distance of 47m from the ground. Find the length of the cable

If you are assuming a parabola, then a little work will show that the equation is

y = ax^2+bx+c
2500a-50b+c = 50
2500a+50b+c = 70
a(-b/2a)^2 + b(-b/2a) + c = 47

y = 1/200 x^2 - 1/5 x + 47
and the vertex is at (20,47)

Now, the length of cable is thus

s = ∫[-50,50] √(1+(x/100-1/5)^2) dx
= 105.8

The curve of a freely hanging cable of uniform density, however, is not a parabola, but a catenary of the form

y = a cosh((x-h)/a)

This gets rather complicated, and the parabola is a good approximation anyway, so let's stop here.

To find the length of the cable, we can use the Pythagorean theorem.

Let's consider the right triangle formed by the height of the first tower, the distance from the ground to the lowest part of the cable, and the length of the cable. The length of the cable represents the hypotenuse of this triangle.

Using the Pythagorean theorem, we can find the length of the cable:

c^2 = a^2 + b^2

Where:
c = length of the cable
a = height of the first tower
b = distance from the ground to the lowest part of the cable

Substituting the given values into the equation, we have:

c^2 = 50^2 + 47^2
c^2 = 2500 + 2209
c^2 = 4709

Taking the square root of both sides to solve for c, we have:

c ≈ √4709
c ≈ 68.56

Therefore, the length of the cable is approximately 68.56 meters.

To find the length of the cable, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, we have a right triangle formed by the cable, the vertical distance from the ground to the lowest part of the cable, and the horizontal distance between the two towers. Let's call the length of the cable "c", the height of the first tower "a", the height of the second tower "b", and the horizontal distance between the two towers "x".

Using the Pythagorean theorem, we can set up the following equation:

c^2 = x^2 + (a - b)^2

In our case, x is given as 100 meters, a is given as 50 meters, b is given as 70 meters, and the vertical distance from the ground to the lowest part of the cable is 47 meters.

Plugging in the values, we can solve for c:

c^2 = 100^2 + (50 - 70)^2
c^2 = 100^2 + (-20)^2
c^2 = 10000 + 400
c^2 = 10400
c ≈ √10400
c ≈ 101.98 meters

Therefore, the length of the cable is approximately 101.98 meters.